1419:
1478:
1407:
1462:
7834:
1490:
2183:
6643:
140:
7822:(closely related to the Collatz conjecture) is false. Hence if BB(15) was known, and this machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples exist (which proves the conjecture true). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(15) will be very hard to compute, at least as difficult as settling this Collatz-like conjecture.
180:
156:
132:
59:
6520:
172:
5236:
6561:
4398:
1151:
6258:
1588:.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence. The argument is not a proof because it assumes that Hailstone sequences are assembled from uncorrelated probabilistic events. (It does rigorously establish that the
1418:
4282:
93:, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. The conjecture has been shown to hold for all positive integers up to
2385:
4779:
3936:
The
Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms. The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the
7735:
Modifying the condition in this way can make a problem either harder or easier to solve (intuitively, it is harder to justify a positive answer but might be easier to justify a negative one). Kurtz and Simon proved that the universally quantified problem is, in fact, undecidable and even higher in
2646:
1817:
836:
4932:
3673:
3204:
2128:
contiguous subsequences, each consisting of an increasing sequence of odd numbers, followed by a decreasing sequence of even numbers. For instance, if the cycle consists of a single increasing sequence of odd numbers followed by a decreasing sequence of even numbers, it is called a
122:
stated in 2010 that the
Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics". However, though the Collatz conjecture itself remains open, efforts to solve the problem have led to new techniques and many partial results.
108:, who introduced the idea in 1937, two years after receiving his doctorate. The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like
6515:{\displaystyle {\begin{aligned}f(z)\triangleq \;&{\frac {z}{2}}\cos ^{2}\left({\frac {\pi }{2}}z\right)+{\frac {3z+1}{2}}\sin ^{2}\left({\frac {\pi }{2}}z\right)\,+\\&{\frac {1}{\pi }}\left({\frac {1}{2}}-\cos(\pi z)\right)\sin(\pi z)+h(z)\sin ^{2}(\pi z)\end{aligned}}}
3756:
An extension to the
Collatz conjecture is to include all integers, not just positive integers. Leaving aside the cycle 0 β 0 which cannot be entered from outside, there are a total of four known cycles, which all nonzero integers seem to eventually fall into under iteration of
355:
1461:
595:
If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found.
7052:
For the special purpose of searching for a counterexample to the
Collatz conjecture, this precomputation leads to an even more important acceleration, used by TomΓ‘s Oliveira e Silva in his computational confirmations of the Collatz conjecture up to large values
472:
5630:
5539:
2215:
4625:
4561:
4277:
2491:
1670:
689:
4791:
5748:
4393:{\displaystyle {\frac {151}{47}}\rightarrow {\frac {250}{47}}\rightarrow {\frac {125}{47}}\rightarrow {\frac {211}{47}}\rightarrow {\frac {340}{47}}\rightarrow {\frac {170}{47}}\rightarrow {\frac {85}{47}}\rightarrow {\frac {151}{47}}.}
6760:
6164:
3076:
5226:
4107:
2168:
values may be ruled out. To state the argument more intuitively; we do not have to search for cycles that have less than 92 subsequences, where each subsequence consists of consecutive ups followed by consecutive downs.
5906:
5829:
5117:
216:
7548:
3548:
1631:(in the sense of logarithmic density) Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. Responding to this work,
1608:, almost every positive integer has a finite stopping time. In other words, almost every Collatz sequence reaches a point that is strictly below its initial value. The proof is based on the distribution of
1543:
on the length of the cycle, can be proven based on the value of the lowest term in the cycle. Therefore, computer searches to rule out cycles that have a small lowest term can strengthen these constraints.
1205:
0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106, 5, 26, 13, 13, 21, 21, 21, 34, 8, 109, 8, 29, 16, 16, 16, 104, 11, 24, 24, ... (sequence
3371:. This implies that every number is uniquely identified by its parity sequence, and moreover that if there are multiple Hailstone cycles, then their corresponding parity cycles must be different.
6263:
1505:
Although the conjecture has not been proven, most mathematicians who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it.
365:
7994:
The problem is also known by several other names, including: Ulam's conjecture, the
Hailstone problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and the Collatz problem.
6852:
2057:
4413:
1406:
6802:
4132:
1477:
3937:
integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added. A closely related fact is that the
Collatz map extends to the ring of
6036:
6217:
3948:
is generated by exactly one rational. Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity
Conjecture).
1489:
8545:
6605:
5160:
5013:
4961:
5545:
5454:
1980:
9290:
2923:, where each repetend is optionally rotated and then replicated up to a finite number of bits. It is only in binary that this occurs. Conjecturally, every binary string
5168:
8402:
5368:
6003:
590:
5641:
7792:
5446:
5419:
6246:
5325:
5276:
6660:
5018:
2380:{\displaystyle R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1,2,3,5\\\left\{2n,{\frac {n-1}{3}}\right\}&{\text{if }}n\equiv 4\end{cases}}{\pmod {6}}.}
1183:
1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, ... (sequence
6633:
6547:
6059:
5970:
5946:
5388:
5296:
4774:{\displaystyle T_{d}(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}},\\{\frac {3x+d}{2}}&{\text{if }}x\equiv 1{\pmod {2}}.\end{cases}}}
557:
537:
7458:
359:
Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next.
6075:
2389:
So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads backwards to all positive integers. For any integer
2641:{\displaystyle R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1\\\left\{2n,{\frac {2n-1}{3}}\right\}&{\text{if }}n\equiv 2\end{cases}}{\pmod {3}}.}
1812:{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}
831:{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}
9445:
4927:{\displaystyle T(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}}\\{\frac {3x+1}{2}}&{\text{if }}x\equiv 1{\pmod {2}}\end{cases}}}
2714:. Conjecturally, this inverse relation forms a tree except for a 1β2 loop (the inverse of the 1β2 loop of the function f(n) revised as indicated above).
9318:
by TomΓ‘s
Oliveira e Silva continues to verify the Collatz conjecture (with fewer statistics than Eric Roosendaal's page but with further progress made).
8487:
3986:
7270:
4611:
of a rational is not a multiple of 3, then all the iterates have the same denominator and the sequence of numerators can be obtained by applying the "
1347:
1213:
1190:
1168:
1144:
7169:
for which such an inequality holds, so checking the
Collatz conjecture for one starting value is as good as checking an entire congruence class. As
2140:. Simons (2005) used Steiner's method to prove that there is no 2-cycle. Simons and de Weger (2005) extended this proof up to 68-cycles; there is no
78:
52:
9150:
Proceedings of the 4th
International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007
3491:. This allows one to predict that certain forms of numbers will always lead to a smaller number after a certain number of iterations: for example,
6572:
Most of the points have orbits that diverge to infinity. Coloring these points based on how fast they diverge produces the image on the left, for
5916:, further investigated by Marc Chamberland. He showed that the conjecture does not hold for positive real numbers since there are infinitely many
5834:
5757:
3668:{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\{\frac {3n+1}{2}}&{\text{if }}n\equiv 1.\end{cases}}{\pmod {2}}}
3199:{\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\{\frac {3n+1}{2}}&{\text{if }}n\equiv 1\end{cases}}{\pmod {2}}.}
9165:
8222:
838:
This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process.
9415:
1354:
These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example,
7181:. Only an exponentially small fraction of the residues survive. For example, the only surviving residues mod 32 are 7, 15, 27, and 31.
6608:
6550:
879:, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1.
8175:
8139:
8105:
8025:
7987:
4968:
1471:
values are shown. The first thick line towards the middle of the plot corresponds to the tip at 27, which reaches a maximum at 9232.
1161:
1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, ... (sequence
350:{\displaystyle f(n)={\begin{cases}n/2&{\text{if }}n\equiv 0{\pmod {2}},\\3n+1&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}
2015:
8620:
7025:(0...31, 5) = {β―0, 2, 1, 1, 2, 2, 2, 20, 1, 26, 1, 10, 4, 4, 13, 40, 2, 5, 17, 17, 2, 2, 20, 20, 8, 22, 8, 71, 26, 26, 80, 242β―}.
69:
of small numbers under the Collatz map, skipping even numbers. The Collatz conjecture states that all paths eventually lead to 1.
8980:
7869:
3768:
Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first.
9185:
Ben-Amram, Amir M. (2015). "Mortality of iterated piecewise affine functions over the integers: Decidability and complexity".
9357:
518:
This process will eventually reach the number 1, regardless of which positive integer is chosen initially. That is, for each
8059:
According to Lagarias (1985), p. 4, the name "Syracuse problem" was proposed by Hasse in the 1950s, during a visit to
8214:
7795:
6525:
is an interpolation of the Collatz map to the complex plane. The reason for adding the extra term is to make all integers
7810:
Collatz and related conjectures are often used when studying computation complexity. The connection is made through the
1649:, Krasikov and Lagarias showed that the number of integers in the interval that eventually reach 1 is at least equal to
9435:
6526:
2206:
2189:
1624:
1383:
The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the
2927:
that ends with a '1' can be reached by a representation of this form (where we may add or delete leading '0's to
1605:
9107:
8859:"Working in binary protects the repetends of 1/3: Comment on Colussi's 'The convergence classes of Collatz function'"
7012:, one can jump ahead 5 steps on each iteration by separating out the 5 least significant bits of a number and using
9440:
9297:
6807:
1528:
may be found when considering very large (or possibly immense) positive integers, as in the case of the disproven
1157:
Numbers with a total stopping time longer than that of any smaller starting value form a sequence beginning with:
7019:(0...31, 5) = {β―0, 3, 2, 2, 2, 2, 2, 4, 1, 4, 1, 3, 2, 2, 3, 4, 1, 2, 3, 3, 1, 1, 3, 3, 2, 3, 2, 4, 3, 3, 4, 5β―},
6769:
5917:
2944:
2839:
2197:
There is another approach to prove the conjecture, which considers the bottom-up method of growing the so-called
7184:
Integers divisible by 3 cannot form a cycle, so these integers do not need to be checked as counter examples.
1539:
However, such verifications may have other implications. Certain constraints on any non-trivial cycle, such as
7041:
6008:
6069:
9286:
by David BaΕina verifies Convergence of the Collatz conjecture for large values. (furthest progress so far)
4815:
4656:
3572:
3100:
2515:
2239:
1694:
1524:
This computer evidence is still not rigorous proof that the conjecture is true for all starting values, as
713:
467:{\displaystyle a_{i}={\begin{cases}n&{\text{for }}i=0,\\f(a_{i-1})&{\text{for }}i>0\end{cases}}}
240:
9279:
8125:
7737:
6168:
3730:, and iteration of the tag operation halts on any word of length less than 2. (Adapted from De Mol.)
2210:
1637:
wrote that Tao "came away with one of the most significant results on the Collatz conjecture in decades".
207:
81:. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every
8559:
8539:
5625:{\displaystyle g_{2}(n)={\begin{cases}0,&n{\text{ is even,}}\\1,&n{\text{ is odd,}}\end{cases}}}
5534:{\displaystyle g_{1}(n)={\begin{cases}1,&n{\text{ is even,}}\\0,&n{\text{ is odd,}}\end{cases}}}
1613:
7833:
9072:
6575:
5136:
4989:
4937:
4129:
has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction
869:
takes longer to reach 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 .
9430:
8788:
8669:
8510:
6651:
4569:, that is, not partitionable into identical sub-cycles. As an illustration of this, the parity cycle
4556:{\displaystyle {\frac {3^{3}2^{1}+3^{2}2^{2}+3^{1}2^{5}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {250}{47}}.}
1646:
90:
5576:
5485:
4272:{\displaystyle {\frac {3^{3}2^{0}+3^{2}2^{2}+3^{1}2^{3}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {151}{47}}}
3733:
The Collatz conjecture equivalently states that this tag system, with an arbitrary finite string of
1952:
1432:
axis represents the highest number reached during the chain to 1. This plot shows a restricted
657:
doesn't exist, we say that the stopping time or the total stopping time, respectively, is infinite.
387:
9273:
8074:
8060:
7955:
7859:
7449:
2440:
1529:
8715:
2193:
generated in bottom-up fashion. The graph includes all numbers with an orbit length of 21 or less.
9394:
9376:
8952:
8693:
8659:
8421:
8371:
8262:
8121:
8042:
7864:
7839:
7445:
5925:
2086:
1533:
203:
6557:. They conjectured that the latter is not the case, which would make all integer orbits finite.
8097:
7847:
6854:. The corresponding Julia set, shown on the right, consists of uncountably many curves, called
5743:{\displaystyle f(x)\triangleq {\frac {x}{2}}\cdot g_{1}(x)\,+\,{\frac {3x+1}{2}}\cdot g_{2}(x)}
9323:
9161:
9145:
8999:
8913:
8282:
8218:
8171:
8159:
8135:
8101:
7983:
7923:
6642:
5330:
5242:
of the orbit 10 β 5 β 8 β 4 β 2 β 1 β ... in an extension of the Collatz map to the real line.
5129:. Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that
2902:
1820:
8131:
5975:
3978:, then the unique rational which generates immediately and periodically this parity cycle is
1556:
numbers in the sequence generated by the Collatz process, then each odd number is on average
562:
9402:
9221:
9194:
9153:
9053:
8989:
8944:
8903:
8870:
8837:
8796:
8677:
8639:
8591:
8518:
8461:
8411:
8363:
8313:
8272:
8228:
8202:
8181:
8089:
8034:
7361:
6039:
5921:
5913:
2940:
188:
139:
119:
86:
82:
66:
8810:
8757:
8689:
8605:
7770:
6755:{\displaystyle f(z)\triangleq {\frac {z}{2}}+{\frac {1}{4}}(2z+1)\left(1-e^{i\pi z}\right)}
5424:
5397:
3924:
The generalized Collatz conjecture is the assertion that every integer, under iteration by
9301:
9283:
8932:
8806:
8753:
8685:
8601:
8232:
8185:
8167:
7931:
7818:
that halts. There is a 15 state Turing machine that halts if and only if a conjecture by
7691:
6249:
6222:
5301:
4964:
3938:
2182:
1633:
1592:
extension of the Collatz process has two division steps for every multiplication step for
1525:
155:
89:
in which each term is obtained from the previous term as follows: if the previous term is
5253:
159:
Histogram of total stopping times for the numbers 1 to 10. Total stopping time is on the
143:
Histogram of total stopping times for the numbers 1 to 10. Total stopping time is on the
8792:
8673:
8514:
6650:
There are many other ways to define a complex interpolating function, such as using the
6159:{\displaystyle {\tfrac {1}{\pi }}\left({\tfrac {1}{2}}-\cos(\pi z)\right)\sin(\pi z)\,+}
4604:
are the only parity cycles generated by positive whole numbers (1 and 2, respectively).
3333:, depends on the parity. The parity sequence is the same as the sequence of operations.
9384:
9366:
8155:
7939:
7815:
7033:
6618:
6532:
6044:
5955:
5931:
5373:
5281:
4972:
3944:
When using the "shortcut" definition of the Collatz map, it is known that any periodic
2951:. The machine will perform the following three steps on any odd number until only one
2400:
1589:
542:
522:
105:
62:
8576:
8416:
8393:
3761:. These cycles are listed here, starting with the well-known cycle for positive
1572:
of the previous one. (More precisely, the geometric mean of the ratios of outcomes is
131:
58:
9424:
8466:
8445:
8375:
8317:
8090:
7819:
6554:
6065:
5391:
3435:
there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is
2831:
118:
said about the Collatz conjecture: "Mathematics may not be ready for such problems."
115:
9326:
9173:
8956:
8697:
1179:
trajectory point is greater than that of any smaller starting value are as follows:
9057:
8948:
8038:
7947:
7752:-complete. This hardness result holds even if one restricts the class of functions
6252:. Since this expression evaluates to zero for real integers, the extended function
2443:
except for the 1β2β4 loop (the inverse of the 4β2β1 loop of the unaltered function
1384:
9352:
8801:
8772:
5221:{\displaystyle Q\left(\mathbb {Z} ^{+}\right)\subset {\tfrac {1}{3}}\mathbb {Z} .}
1495:
The number of iterations it takes to get to one for the first 100 million numbers.
9157:
4102:{\displaystyle {\frac {3^{m-1}2^{k_{0}}+\cdots +3^{0}2^{k_{m-1}}}{2^{n}-3^{m}}}.}
3228:, then we can define the Collatz parity sequence (or parity vector) for a number
9311:
by Eric Roosendaal verifies the Collatz conjecture for larger and larger values.
7811:
5239:
3531:. Whether those smaller numbers continue to 1, however, depends on the value of
1620:
1540:
171:
17:
8367:
7448:
proved that a natural generalization of the Collatz problem is algorithmically
6553:, which implies that any integer is either eventually periodic or belongs to a
3427:
is how many increases were encountered during that sequence. For example, for
1949:
without shortcut). If it can be shown that for all positive integers less than
9406:
9388:
9370:
9344:
8875:
8858:
8842:
8825:
8643:
7829:
6560:
5909:
5130:
4596:
In this context, assuming the validity of the Collatz conjecture implies that
3739:
3678:
2808:
2787:
1628:
1593:
1513:
The conjecture has been checked by computer for all starting values up to 2 β
109:
9003:
8917:
8286:
9331:
9212:
Michel, Pascal (1993). "Busy beaver competition and Collatz-like problems".
8596:
8332:
6612:
5949:
5901:{\displaystyle g_{2}(x)\triangleq \sin ^{2}\left({\tfrac {\pi }{2}}x\right)}
5824:{\displaystyle g_{1}(x)\triangleq \cos ^{2}\left({\tfrac {\pi }{2}}x\right)}
5247:
5235:
2800:
8994:
8971:
5112:{\displaystyle Q(x)=\sum _{k=0}^{\infty }\left(T^{k}(x)\mod 2\right)2^{k}.}
3941:, which contains the ring of rationals with odd denominators as a subring.
859:
without "shortcut", one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.
183:
Total stopping time of numbers up to 250, 1000, 4000, 20000, 100000, 500000
9308:
8908:
7814:
function, where BB(n) is the maximum number of steps taken by any n state
4971:
with respect to the 2-adic measure. Moreover, its dynamics is known to be
8354:
Barina, David (2020). "Convergence verification of the Collatz problem".
7799:
7543:{\displaystyle {g(n)=a_{i}n+b_{i}}{\text{ when }}{n\equiv i{\pmod {P}}},}
7439:
6889:
function from that section), break up the current number into two parts,
6881:
above gives a way to speed up simulation of the sequence. To jump ahead
5126:
3928:, eventually falls into one of the four cycles above or the cycle 0 β 0.
3451:
there is only 1 increase as 1 rises to 2 and falls to 1 so the result is
686:
is odd, one may instead use the "shortcut" form of the Collatz function:
9340:
9294:
179:
9225:
9198:
8425:
8277:
8250:
8077:. St Andrews University School of Mathematics and Statistics, Scotland.
8046:
6636:
6565:
2939:
Repeated applications of the Collatz function can be represented as an
1176:
31:
9239:
8681:
8523:
7005:
is the number of odd numbers encountered on the way. For example, if
2151:. Hercher extended the method further and proved that there exists no
2136:
Steiner (1977) proved that there is no 1-cycle other than the trivial
1220:
The starting value having the largest total stopping time while being
1149:
8664:
4406:
is associated to one of the above fractions. For instance, the cycle
3073:
For this section, consider the shortcut form of the Collatz function
660:
The Collatz conjecture asserts that the total stopping time of every
9416:
Are computers ready to solve this notoriously unwieldy math problem?
9257:
7404:
The Collatz conjecture is equivalent to the statement that, for all
7165:
which is not a counterexample to the Collatz conjecture, there is a
1150:
9315:
8750:
Proceedings of the 7th Manitoba Conference on Numerical Mathematics
8267:
8251:"Almost all orbits of the Collatz map attain almost bounded values"
6901:(the rest of the bits as an integer). The result of jumping ahead
6641:
6559:
5234:
2181:
2008:
without shortcut). In fact, Eliahou (1993) proved that the period
178:
170:
154:
138:
135:
Numbers from 1 to 9999 and their corresponding total stopping time
130:
57:
7597:
is always an integer. The standard Collatz function is given by
7036:
and storage to speed up the resulting calculation by a factor of
6766:
which exhibit different dynamics. In this case, for instance, if
1667:
In this part, consider the shortcut form of the Collatz function
6038:. Moreover, the set of unbounded orbits is conjectured to be of
9276:
8891:
8481:
8479:
8477:
49:
With enough repetition, do all positive integers converge to 1?
9040:
Letherman, Simon; Schleicher, Dierk; Wood, Reg (1999). "The (3
7117:. For instance, the first counterexample must be odd because
2948:
2879:
is a finite and contiguous extract from the representation of
1982:
the Collatz sequences reach 1, then this bound would raise to
8748:
Steiner, R. P. (1977). "A theorem on the syracuse problem".
5165:
An equivalent formulation of the Collatz conjecture is that
3347:, it can be shown that the parity sequences for two numbers
2997:
s (that is, repeatedly divide by 2 until the result is odd).
1412:
Directed graph showing the orbits of the first 1000 numbers.
7265:
5618:
5527:
5394:
function. A simple way to do this is to pick two functions
4920:
4767:
3645:
3173:
2974:
Add this to the original number by binary addition (giving
2615:
2354:
1805:
1467:
The same plot as the previous one but on log scale, so all
1342:
1208:
1185:
1163:
1139:
824:
460:
343:
9017:
Chamberland, Marc (1996). "A continuous extension of the 3
8450:+ 1 problem: new lower bounds on nontrivial cycle lengths"
6064:
Letherman, Schleicher, and Wood extended the study to the
4565:
For a one-to-one correspondence, a parity cycle should be
1923:
The length of a non-trivial cycle is known to be at least
8621:"Mathematician Proves Huge Result on 'Dangerous' Problem"
7173:
increases, the search only needs to check those residues
7109:, then the first counterexample, if it exists, cannot be
2466:
is replaced by the common substitute "shortcut" relation
8344:(Note: "Delay records" are total stopping time records.)
6897:
least significant bits, interpreted as an integer), and
2488:, the Collatz graph is defined by the inverse relation,
9304:
that verifies the Collatz conjecture for larger values.
9146:"The undecidability of the generalized Collatz problem"
9131:
Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder
9077:+ 1 problem by means of the COMETA grid infrastructure"
6607:. The inner black regions and the outer region are the
1483:
The tree of all the numbers having fewer than 20 steps.
6833:
6097:
6080:
5879:
5802:
5199:
2811:
to odd numbers. Now suppose that for some odd number
664:
is finite. It is also equivalent to saying that every
8648: + 1 problem using difference inequalities"
8300:
Leavens, Gary T.; Vermeulen, Mike (December 1992). "3
7773:
7461:
6810:
6772:
6663:
6621:
6578:
6568:
centered at the origin, with real parts from -5 to 5.
6535:
6261:
6225:
6171:
6078:
6047:
6011:
5978:
5958:
5934:
5837:
5760:
5644:
5548:
5457:
5427:
5400:
5376:
5333:
5304:
5284:
5256:
5171:
5139:
5021:
4992:
4940:
4794:
4628:
4416:
4285:
4135:
3989:
3551:
3079:
2494:
2218:
2018:
1955:
1673:
692:
565:
545:
525:
368:
219:
9397:
from the original on 2021-12-11 – via YouTube.
9379:
from the original on 2021-12-11 – via YouTube.
9129:
Conway, John H. (1972). "Unpredictable iterations".
9044: + 1)-problem and holomorphic dynamics".
8970:Bernstein, Daniel J.; Lagarias, Jeffrey C. (1996).
2963:to the (right) end of the number in binary (giving
2162:. As exhaustive computer searches continue, larger
1623:improved this result by showing, using logarithmic
8577:"A stopping time problem on the positive integers"
7786:
7542:
7455:Specifically, he considered functions of the form
6846:
6796:
6754:
6627:
6599:
6541:
6514:
6240:
6211:
6158:
6053:
6030:
5997:
5964:
5940:
5900:
5823:
5742:
5624:
5533:
5440:
5413:
5382:
5362:
5319:
5290:
5270:
5220:
5154:
5111:
5007:
4955:
4926:
4773:
4555:
4392:
4271:
4101:
3667:
3198:
2640:
2379:
2051:
1974:
1811:
830:
584:
551:
531:
466:
349:
8933:"Embedding the 3x+1 Conjecture in a 3x+d Context"
8892:"The set of rational cycles for the 3x+1 problem"
8709:
8707:
8439:
8437:
8435:
8197:
8195:
8014:
8012:
8010:
8008:
8006:
8004:
8002:
5635:and use them as switches for our desired values:
187:Consider the following operation on an arbitrary
9148:. In Cai, J.-Y.; Cooper, S. B.; Zhu, H. (eds.).
8570:
8568:
8544:: CS1 maint: bot: original URL status unknown (
8403:Proceedings of the American Mathematical Society
8387:
8385:
7978:Maddux, Cleborne D.; Johnson, D. Lamont (1997).
7246:of positive odd integers into itself, for which
7177:that are not eliminated by lower values of
3006:The starting number 7 is written in base two as
2935:As an abstract machine that computes in base two
7767:in a simplified version of this form, with all
7697:Closer to the Collatz problem is the following
6639:pattern, sometimes called a "Collatz fractal".
6549:. With this, they show that no integer is in a
4622: " generalization of the Collatz function
2439:. Conjecturally, this inverse relation forms a
2122:-cycle is a cycle that can be partitioned into
9411:(short video). Veritasium – via YouTube.
7586:are rational numbers which are so chosen that
7276:Some properties of the Syracuse function are:
3543:For the Collatz function in the shortcut form
9108:"The Long Search for Collatz Counterexamples"
8931:Belaga, Edward G.; Mignotte, Maurice (1998).
8826:"The convergence classes of Collatz function"
8306:Computers & Mathematics with Applications
6847:{\displaystyle f(z)\approx z+{\tfrac {1}{4}}}
2448:
2052:{\displaystyle p=301994a+17087915b+85137581c}
8:
8096:. Oxford: Oxford University Press. pp.
5250:by choosing any function which evaluates to
3932:Iterating on rationals with odd denominators
2527:
2518:
2251:
2242:
198:If the number is odd, triple it and add one.
9084:Grid Open Days at the University of Palermo
6797:{\displaystyle \operatorname {Im} (z)\gg 1}
6646:Julia set of the exponential interpolation.
3737:as the initial word, eventually halts (see
3677:Hailstone sequences can be computed by the
1230:less than 1000 is 871, which has 178 steps,
9408:The simplest math problem no one can solve
9021: + 1 problem to the real line".
8773:"On the nonexistence of 2-cycles for the 3
8527:. Archived from the original on 2022-03-18
8488:"Theoretical and computational bounds for
6281:
4279:as the latter leads to the rational cycle
1521:. All values tested so far converge to 1.
1395:times to reach 1, and is never increased.
1233:less than 10 is 6171, which has 261 steps,
8993:
8907:
8874:
8841:
8800:
8663:
8595:
8522:
8465:
8415:
8276:
8266:
7778:
7772:
7520:
7510:
7505:
7498:
7482:
7462:
7460:
6877:
6832:
6809:
6771:
6735:
6692:
6679:
6662:
6620:
6577:
6534:
6487:
6414:
6399:
6390:
6372:
6358:
6333:
6312:
6298:
6284:
6262:
6260:
6224:
6188:
6170:
6152:
6096:
6079:
6077:
6046:
6021:
6010:
5988:
5977:
5957:
5933:
5878:
5864:
5842:
5836:
5801:
5787:
5765:
5759:
5725:
5697:
5696:
5692:
5677:
5660:
5643:
5610:
5590:
5571:
5553:
5547:
5519:
5499:
5480:
5462:
5456:
5432:
5426:
5405:
5399:
5375:
5352:
5332:
5303:
5283:
5260:
5255:
5211:
5210:
5198:
5185:
5181:
5180:
5170:
5146:
5142:
5141:
5138:
5100:
5087:
5086:
5067:
5052:
5041:
5020:
4999:
4995:
4994:
4991:
4947:
4943:
4942:
4939:
4901:
4887:
4864:
4844:
4830:
4818:
4810:
4793:
4745:
4731:
4708:
4685:
4671:
4659:
4651:
4633:
4627:
4540:
4528:
4515:
4503:
4493:
4480:
4470:
4457:
4447:
4434:
4424:
4417:
4415:
4377:
4364:
4351:
4338:
4325:
4312:
4299:
4286:
4284:
4259:
4247:
4234:
4222:
4212:
4199:
4189:
4176:
4166:
4153:
4143:
4136:
4134:
4087:
4074:
4054:
4049:
4039:
4018:
4013:
3997:
3990:
3988:
3945:
3649:
3628:
3605:
3587:
3575:
3567:
3550:
3177:
3156:
3133:
3115:
3103:
3095:
3078:
2619:
2598:
2570:
2532:
2510:
2493:
2358:
2337:
2312:
2256:
2234:
2217:
2017:
1966:
1954:
1783:
1769:
1746:
1723:
1709:
1697:
1689:
1672:
1609:
1227:less than 100 is 97, which has 118 steps,
802:
788:
765:
742:
728:
716:
708:
691:
570:
564:
544:
524:
443:
426:
395:
382:
373:
367:
321:
307:
270:
256:
246:
235:
218:
45:For odd numbers, multiply by 3 and add 1.
8073:O'Connor, J.J.; Robertson, E.F. (2006).
7982:. New York: Haworth Press. p. 160.
3770:
2012:of any non-trivial cycle is of the form
1440:values produce intermediates as high as
195:If the number is even, divide it by two.
9144:Kurtz, Stuart A.; Simon, Janos (2007).
7970:
7881:
6611:, and the boundary between them is the
6068:. They used Chamberland's function for
6031:{\displaystyle (1.1925...;\,2.1386...)}
5246:The Collatz map can be extended to the
2838:can be written as the concatenation of
1915:of period 2, called the trivial cycle.
1402:
101:, but no general proof has been found.
53:(more unsolved problems in mathematics)
9240:"Hardness of busy beaver value BB(15)"
9023:Dynam. Contin. Discrete Impuls Systems
8537:
8244:
8242:
8023:+ 1 problem and its generalizations".
6955:can be precalculated for all possible
3010:. The resulting Collatz sequence is:
1224:less than 10 is 9, which has 19 steps,
27:Open problem on 3x+1 and x/2 functions
8619:Hartnett, Kevin (December 11, 2019).
5390:is an odd integer. This is called an
3718:In this system, the positive integer
3487:; it depends only on the behavior of
1428:axis represents starting number, the
175:Iteration time for inputs of 2 to 10.
112:in a cloud), or as wondrous numbers.
7:
9314:Another ongoing volunteer computing
8857:Hew, Patrick Chisan (7 March 2016).
7794:equal to zero, are formalized in an
7690:is undecidable, by representing the
6212:{\displaystyle h(z)\sin ^{2}(\pi z)}
5231:Iterating on real or complex numbers
4577:are associated to the same fraction
3980:
2819:times yields the number 1 (that is,
2173:Other formulations of the conjecture
1848:of distinct positive integers where
104:It is named after the mathematician
9372:Uncrackable? The Collatz conjecture
9073:"Looking for class records in the 3
8824:Colussi, Livio (9 September 2011).
8019:Lagarias, Jeffrey C. (1985). "The 3
7528:
6885:steps on each iteration (using the
4909:
4902:
4852:
4845:
4753:
4746:
4693:
4686:
3657:
3212:is the parity of a number, that is
3185:
2627:
2366:
1791:
1784:
1731:
1724:
810:
803:
750:
743:
329:
322:
278:
271:
9446:Unsolved problems in number theory
8164:Unsolved Problems in Number Theory
8130:. New York: Basic Books. pp.
5370:(for the "shortcut" version) when
5298:is an even integer, and to either
5053:
1340:, which has 1348 steps. (sequence
25:
8486:Simons, J.; de Weger, B. (2005).
8417:10.1090/S0002-9939-1981-0603593-2
8026:The American Mathematical Monthly
7659:. Conway proved that the problem
7138:; and it must be 3 mod 4 because
3955:and includes odd numbers exactly
7832:
7709:, does the sequence of iterates
7671:, does the sequence of iterates
6600:{\displaystyle h(z)\triangleq 0}
5155:{\displaystyle \mathbb {Z} _{2}}
5008:{\displaystyle \mathbb {Z} _{2}}
4956:{\displaystyle \mathbb {Z} _{2}}
1488:
1476:
1460:
1417:
1405:
79:unsolved problems in mathematics
9390:Uncrackable? Collatz conjecture
9307:An ongoing volunteer computing
8981:Canadian Journal of Mathematics
7870:Residue-class-wise affine group
7521:
5082:
3650:
3483:is independent of the value of
3178:
2620:
2359:
34:Unsolved problem in mathematics
9393:(extra footage). Numberphile.
9358:Wolfram Demonstrations Project
9214:Archive for Mathematical Logic
9058:10.1080/10586458.1999.10504402
8949:10.1080/10586458.1998.10504364
8088:Pickover, Clifford A. (2001).
8039:10.1080/00029890.1985.11971528
7532:
7522:
7472:
7466:
6978:is the result of applying the
6820:
6814:
6785:
6779:
6717:
6702:
6673:
6667:
6588:
6582:
6505:
6496:
6480:
6474:
6465:
6456:
6442:
6433:
6275:
6269:
6235:
6229:
6206:
6197:
6181:
6175:
6149:
6140:
6126:
6117:
6025:
6012:
5992:
5979:
5854:
5848:
5777:
5771:
5737:
5731:
5689:
5683:
5654:
5648:
5565:
5559:
5474:
5468:
5349:
5334:
5079:
5073:
5031:
5025:
4913:
4903:
4856:
4846:
4804:
4798:
4757:
4747:
4697:
4687:
4645:
4639:
4593:when reduced to lowest terms.
4374:
4361:
4348:
4335:
4322:
4309:
4296:
4125:For example, the parity cycle
3722:is represented by a string of
3661:
3651:
3561:
3555:
3411:is the result of applying the
3291:Which operation is performed,
3189:
3179:
3089:
3083:
2631:
2621:
2504:
2498:
2370:
2360:
2228:
2222:
2085:. This result is based on the
1975:{\displaystyle 3\times 2^{69}}
1795:
1785:
1735:
1725:
1683:
1677:
814:
804:
754:
744:
702:
696:
438:
419:
333:
323:
282:
272:
229:
223:
42:For even numbers, divide by 2;
1:
8802:10.1090/s0025-5718-04-01728-4
8356:The Journal of Supercomputing
8215:American Mathematical Society
8207:The Ultimate Challenge: The 3
7796:esoteric programming language
4967:, where it is continuous and
3951:If a parity cycle has length
3479:. The power of 3 multiplying
3471:rises and the result will be
9405:(featuring) (30 July 2021).
9375:(short video). Numberphile.
9158:10.1007/978-3-540-72504-6_49
8863:Theoretical Computer Science
8830:Theoretical Computer Science
8731:Journal of Integer Sequences
8467:10.1016/0012-365X(93)90052-U
8318:10.1016/0898-1221(92)90034-F
7328:In more generality: For all
6654:instead of sine and cosine:
5133:trajectories are acyclic in
4934:is well-defined on the ring
4410:is produced by the fraction
3904:β β272 β β136 β β68 β β34 β
3747:Extensions to larger domains
846:For instance, starting with
671:has a finite stopping time.
9096:Lagarias (1985), Theorem D.
7806:In computational complexity
7434:Undecidable generalizations
7362:function iteration notation
7161:. For each starting value
3527:after four applications of
2803:). The resulting function
2717:Alternatively, replace the
2186:The first 21 levels of the
2071:are non-negative integers,
1655:for all sufficiently large
516:The Collatz conjecture is:
9462:
8890:Lagarias, Jeffrey (1990).
8558:Lagarias (1985), section "
8368:10.1007/s11227-020-03368-x
7437:
7412:, there exists an integer
5928:to infinity. The function
4402:Any cyclic permutation of
3507:after two applications of
2815:, applying this operation
2451:section of this article).
1552:If one considers only the
1175:The starting values whose
1080:, 9232, 4616, 2308, 1154,
853:and applying the function
627:. Similarly, the smallest
77:is one of the most famous
9071:Scollo, Giuseppe (2007).
8876:10.1016/j.tcs.2015.12.033
8843:10.1016/j.tcs.2011.05.056
8398: + 1 algorithm"
8160:""E16: The 3x+1 problem""
6072:and added the extra term
3752:Iterating on all integers
1596:2-adic starting values.)
1548:A probabilistic heuristic
9046:Experimental Mathematics
8937:Experimental Mathematics
8771:Simons, John L. (2005).
8444:Eliahou, Shalom (1993).
8392:Garner, Lynn E. (1981).
8255:Forum of Mathematics, Pi
7888:It is also known as the
7196:is an odd integer, then
5363:{\displaystyle (3x+1)/2}
3355:will agree in the first
2897:. The representation of
2449:Statement of the problem
1911:The only known cycle is
1367:has 1132 steps, as does
649:. If one of the indexes
127:Statement of the problem
9106:Clay, Oliver Keatinge.
8597:10.4064/aa-30-3-241-252
8496: + 1 problem"
6070:complex sine and cosine
5998:{\displaystyle (1;\,2)}
4607:If the odd denominator
3743:for a worked example).
2209:defined by the inverse
1324:, which has 1228 steps,
1308:, which has 1132 steps,
1122:, 160, 80, 40, 20, 10,
585:{\displaystyle a_{i}=1}
163:axis, frequency on the
147:axis, frequency on the
9293:) volunteer computing
8995:10.4153/CJM-1996-060-x
8716:"There are no Collatz
8304:+ 1 search programs".
8122:Hofstadter, Douglas R.
7788:
7756:by fixing the modulus
7740:; specifically, it is
7738:arithmetical hierarchy
7699:universally quantified
7544:
7057:. If, for some given
6848:
6798:
6756:
6647:
6629:
6601:
6569:
6543:
6516:
6242:
6213:
6160:
6055:
6032:
5999:
5966:
5942:
5912:of this map lead to a
5902:
5825:
5744:
5626:
5535:
5442:
5415:
5384:
5364:
5321:
5292:
5272:
5243:
5222:
5156:
5113:
5057:
5009:
4957:
4928:
4775:
4557:
4394:
4273:
4103:
3777:Odd-value cycle length
3681:with production rules
3669:
3367:are equivalent modulo
3200:
2642:
2381:
2194:
2053:
1976:
1813:
1292:, which has 986 steps,
1279:, which has 949 steps,
1266:, which has 685 steps,
1253:, which has 524 steps,
1243:, which has 350 steps,
1154:
1098:, 976, 488, 244, 122,
832:
586:
553:
533:
468:
351:
184:
176:
168:
152:
136:
70:
8909:10.4064/aa-56-1-33-53
8575:Terras, Riho (1976).
8560:A heuristic argument"
8249:Tao, Terence (2022).
7980:Logo: A Retrospective
7789:
7787:{\displaystyle b_{i}}
7545:
6849:
6799:
6757:
6645:
6630:
6602:
6563:
6544:
6517:
6243:
6214:
6161:
6056:
6033:
6000:
5967:
5943:
5903:
5826:
5745:
5627:
5536:
5443:
5441:{\displaystyle g_{2}}
5416:
5414:{\displaystyle g_{1}}
5385:
5365:
5322:
5293:
5273:
5238:
5223:
5157:
5114:
5037:
5010:
4958:
4929:
4776:
4558:
4395:
4274:
4104:
3670:
3396:will give the result
3359:terms if and only if
3201:
2643:
2382:
2185:
2054:
1977:
1814:
1614:central limit theorem
1509:Experimental evidence
1153:
833:
587:
554:
534:
469:
352:
206:notation, define the
182:
174:
158:
142:
134:
87:sequences of integers
61:
9152:. pp. 542β553.
8737:(3): Article 23.3.5.
8714:Hercher, C. (2023).
8640:Lagarias, Jeffrey C.
8454:Discrete Mathematics
8333:"3x+1 delay records"
8203:Lagarias, Jeffrey C.
7771:
7459:
7048:Modular restrictions
6878:As a parity sequence
6808:
6770:
6661:
6619:
6576:
6533:
6259:
6241:{\displaystyle h(z)}
6223:
6169:
6076:
6045:
6009:
5976:
5956:
5932:
5835:
5758:
5642:
5546:
5455:
5425:
5398:
5374:
5331:
5320:{\displaystyle 3x+1}
5302:
5282:
5254:
5169:
5137:
5019:
4990:
4938:
4792:
4626:
4414:
4283:
4133:
3987:
3549:
3382:times to the number
3336:Using this form for
3077:
3069:As a parity sequence
2993:Remove all trailing
2901:therefore holds the
2492:
2216:
2016:
1953:
1671:
1647:computer-aided proof
1501:Supporting arguments
1197:Number of steps for
1056:, 7288, 3644, 1822,
690:
563:
543:
523:
366:
217:
85:into 1. It concerns
9436:Arithmetic dynamics
9274:volunteer computing
8793:2005MaCom..74.1565S
8674:2003AcAri.109..237K
8515:2005AcAri.117...51S
8127:GΓΆdel, Escher, Bach
8061:Syracuse University
7956:Syracuse University
7936:Thwaites conjecture
7860:Arithmetic dynamics
7042:spaceβtime tradeoff
6871:Timeβspace tradeoff
6652:complex exponential
5754:One such choice is
5271:{\displaystyle x/2}
3467:then there will be
643:total stopping time
9387:(August 9, 2016).
9324:Weisstein, Eric W.
9300:2017-12-04 at the
9282:2021-08-30 at the
9226:10.1007/BF01409968
9199:10.3233/COM-150032
8976:+ 1 conjugacy map"
8752:. pp. 553β9.
8331:Roosendaal, Eric.
8278:10.1017/fmp.2022.8
8170:. pp. 330β6.
8092:Wonders of Numbers
7928:Kakutani's problem
7865:Modular arithmetic
7840:Mathematics portal
7784:
7540:
7446:John Horton Conway
6844:
6842:
6794:
6752:
6648:
6625:
6597:
6570:
6539:
6512:
6510:
6238:
6209:
6156:
6106:
6089:
6051:
6028:
5995:
5962:
5938:
5898:
5888:
5821:
5811:
5740:
5622:
5617:
5531:
5526:
5438:
5411:
5380:
5360:
5317:
5288:
5268:
5244:
5218:
5208:
5152:
5109:
5005:
4969:measure-preserving
4953:
4924:
4919:
4771:
4766:
4573:and its sub-cycle
4553:
4390:
4269:
4099:
3780:Full cycle length
3665:
3644:
3196:
3172:
2638:
2614:
2454:When the relation
2377:
2353:
2195:
2087:continued fraction
2049:
1972:
1809:
1804:
1534:Mertens conjecture
1155:
990:, 1336, 668, 334,
828:
823:
582:
549:
529:
464:
459:
347:
342:
204:modular arithmetic
185:
177:
169:
153:
137:
75:Collatz conjecture
71:
9441:Integer sequences
9369:(8 August 2016).
9351:Nochella, Jesse.
9327:"Collatz Problem"
9256:Matthews, Keith.
9167:978-3-540-72503-9
9133:. pp. 49β52.
8836:(39): 5409β5419.
8682:10.4064/aa109-3-4
8644:"Bounds for the 3
8524:10.4064/aa117-1-3
8394:"On the Collatz 3
8224:978-0-8218-4940-8
7944:Hasse's algorithm
7508:
7236:Syracuse function
7188:Syracuse function
7065:, the inequality
6841:
6700:
6687:
6628:{\displaystyle f}
6542:{\displaystyle f}
6422:
6407:
6380:
6352:
6320:
6292:
6105:
6088:
6054:{\displaystyle 0}
5965:{\displaystyle 2}
5952:cycles of period
5941:{\displaystyle f}
5887:
5810:
5716:
5668:
5613:
5593:
5522:
5502:
5383:{\displaystyle x}
5291:{\displaystyle x}
5207:
4890:
4883:
4833:
4826:
4734:
4727:
4674:
4667:
4571:(1 1 0 0 1 1 0 0)
4548:
4535:
4385:
4372:
4359:
4346:
4333:
4320:
4307:
4294:
4267:
4254:
4123:
4122:
4094:
3959:times at indices
3922:
3921:
3631:
3624:
3590:
3583:
3159:
3152:
3118:
3111:
2601:
2589:
2535:
2340:
2328:
2259:
1772:
1765:
1712:
1705:
872:The sequence for
791:
784:
731:
724:
682:is even whenever
552:{\displaystyle i}
532:{\displaystyle n}
446:
398:
310:
259:
16:(Redirected from
9453:
9412:
9403:Alex Kontorovich
9398:
9380:
9362:
9337:
9336:
9269:
9265:
9263:
9244:
9243:
9236:
9230:
9229:
9209:
9203:
9202:
9182:
9176:
9171:
9141:
9135:
9134:
9126:
9120:
9119:
9117:
9115:
9103:
9097:
9094:
9088:
9087:
9081:
9068:
9062:
9061:
9037:
9031:
9030:
9014:
9008:
9007:
8997:
8988:(6): 1154β1169.
8967:
8961:
8960:
8928:
8922:
8921:
8911:
8896:Acta Arithmetica
8887:
8881:
8880:
8878:
8854:
8848:
8847:
8845:
8821:
8815:
8814:
8804:
8768:
8762:
8761:
8745:
8739:
8738:
8728:
8711:
8702:
8701:
8667:
8652:Acta Arithmetica
8638:Krasikov, Ilia;
8635:
8629:
8628:
8616:
8610:
8609:
8599:
8584:Acta Arithmetica
8581:
8572:
8563:
8556:
8550:
8549:
8543:
8535:
8533:
8532:
8526:
8503:Acta Arithmetica
8500:
8492:-cycles of the 3
8483:
8472:
8471:
8469:
8441:
8430:
8429:
8419:
8389:
8380:
8379:
8362:(3): 2681β2688.
8351:
8345:
8343:
8341:
8339:
8328:
8322:
8321:
8297:
8291:
8290:
8280:
8270:
8246:
8237:
8236:
8199:
8190:
8189:
8166:(3rd ed.).
8152:
8146:
8145:
8118:
8112:
8111:
8095:
8085:
8079:
8078:
8075:"Lothar Collatz"
8070:
8064:
8057:
8051:
8050:
8016:
7997:
7996:
7975:
7959:
7952:Syracuse problem
7911:
7896:
7886:
7854:
7842:
7837:
7836:
7793:
7791:
7790:
7785:
7783:
7782:
7766:
7759:
7755:
7751:
7750:
7749:
7730:
7723:
7719:
7708:
7685:
7681:
7670:
7666:
7658:
7648:
7638:
7628:
7627:
7625:
7624:
7621:
7618:
7603:
7596:
7585:
7549:
7547:
7546:
7541:
7536:
7535:
7509:
7507: when
7506:
7504:
7503:
7502:
7487:
7486:
7429:
7418:
7411:
7407:
7400:
7399:
7397:
7396:
7393:
7390:
7370:
7359:
7353:
7338:
7334:
7324:
7308:
7289:
7268:
7262:
7261:
7245:
7241:
7238:is the function
7233:
7226:
7225:
7217:
7216:
7203:
7195:
7180:
7176:
7172:
7168:
7164:
7160:
7152:
7137:
7130:
7116:
7112:
7108:
7101:
7064:
7060:
7056:
7039:
7032:
7024:
7018:
7011:
7004:
6989:
6985:
6981:
6977:
6962:
6958:
6954:
6950:
6946:
6938:
6904:
6900:
6896:
6892:
6888:
6884:
6853:
6851:
6850:
6845:
6843:
6834:
6803:
6801:
6800:
6795:
6761:
6759:
6758:
6753:
6751:
6747:
6746:
6745:
6701:
6693:
6688:
6680:
6635:, which forms a
6634:
6632:
6631:
6626:
6609:Fatou components
6606:
6604:
6603:
6598:
6555:wandering domain
6548:
6546:
6545:
6540:
6521:
6519:
6518:
6513:
6511:
6492:
6491:
6449:
6445:
6423:
6415:
6408:
6400:
6397:
6389:
6385:
6381:
6373:
6363:
6362:
6353:
6348:
6334:
6329:
6325:
6321:
6313:
6303:
6302:
6293:
6285:
6247:
6245:
6244:
6239:
6218:
6216:
6215:
6210:
6193:
6192:
6165:
6163:
6162:
6157:
6133:
6129:
6107:
6098:
6090:
6081:
6060:
6058:
6057:
6052:
6037:
6035:
6034:
6029:
6004:
6002:
6001:
5996:
5971:
5969:
5968:
5963:
5947:
5945:
5944:
5939:
5914:dynamical system
5907:
5905:
5904:
5899:
5897:
5893:
5889:
5880:
5869:
5868:
5847:
5846:
5830:
5828:
5827:
5822:
5820:
5816:
5812:
5803:
5792:
5791:
5770:
5769:
5749:
5747:
5746:
5741:
5730:
5729:
5717:
5712:
5698:
5682:
5681:
5669:
5661:
5631:
5629:
5628:
5623:
5621:
5620:
5614:
5611:
5594:
5591:
5558:
5557:
5540:
5538:
5537:
5532:
5530:
5529:
5523:
5520:
5503:
5500:
5467:
5466:
5447:
5445:
5444:
5439:
5437:
5436:
5420:
5418:
5417:
5412:
5410:
5409:
5389:
5387:
5386:
5381:
5369:
5367:
5366:
5361:
5356:
5326:
5324:
5323:
5318:
5297:
5295:
5294:
5289:
5277:
5275:
5274:
5269:
5264:
5227:
5225:
5224:
5219:
5214:
5209:
5200:
5194:
5190:
5189:
5184:
5161:
5159:
5158:
5153:
5151:
5150:
5145:
5124:
5118:
5116:
5115:
5110:
5105:
5104:
5095:
5091:
5072:
5071:
5056:
5051:
5014:
5012:
5011:
5006:
5004:
5003:
4998:
4985:
4962:
4960:
4959:
4954:
4952:
4951:
4946:
4933:
4931:
4930:
4925:
4923:
4922:
4916:
4891:
4888:
4884:
4879:
4865:
4859:
4834:
4831:
4827:
4819:
4784:2-adic extension
4780:
4778:
4777:
4772:
4770:
4769:
4760:
4735:
4732:
4728:
4723:
4709:
4700:
4675:
4672:
4668:
4660:
4638:
4637:
4621:
4610:
4603:
4599:
4592:
4590:
4589:
4586:
4583:
4576:
4572:
4562:
4560:
4559:
4554:
4549:
4541:
4536:
4534:
4533:
4532:
4520:
4519:
4509:
4508:
4507:
4498:
4497:
4485:
4484:
4475:
4474:
4462:
4461:
4452:
4451:
4439:
4438:
4429:
4428:
4418:
4409:
4405:
4399:
4397:
4396:
4391:
4386:
4378:
4373:
4365:
4360:
4352:
4347:
4339:
4334:
4326:
4321:
4313:
4308:
4300:
4295:
4287:
4278:
4276:
4275:
4270:
4268:
4260:
4255:
4253:
4252:
4251:
4239:
4238:
4228:
4227:
4226:
4217:
4216:
4204:
4203:
4194:
4193:
4181:
4180:
4171:
4170:
4158:
4157:
4148:
4147:
4137:
4128:
4117:
4108:
4106:
4105:
4100:
4095:
4093:
4092:
4091:
4079:
4078:
4068:
4067:
4066:
4065:
4064:
4044:
4043:
4025:
4024:
4023:
4022:
4008:
4007:
3991:
3981:
3977:
3958:
3954:
3927:
3771:
3764:
3760:
3736:
3729:
3725:
3721:
3713:
3703:
3693:
3674:
3672:
3671:
3666:
3664:
3648:
3647:
3632:
3629:
3625:
3620:
3606:
3591:
3588:
3584:
3576:
3534:
3530:
3526:
3518:
3510:
3506:
3498:
3490:
3486:
3482:
3478:
3470:
3466:
3462:
3458:
3450:
3442:
3434:
3426:
3422:
3418:
3414:
3410:
3406:
3395:
3381:
3377:
3370:
3366:
3362:
3358:
3354:
3350:
3346:
3332:
3331:
3329:
3328:
3325:
3322:
3312:
3311:
3309:
3308:
3305:
3302:
3287:
3261:
3248:
3231:
3227:
3219:
3211:
3205:
3203:
3202:
3197:
3192:
3176:
3175:
3160:
3157:
3153:
3148:
3134:
3119:
3116:
3112:
3104:
3009:
2996:
2989:
2970:
2962:
2954:
2941:abstract machine
2930:
2926:
2922:
2921:
2919:
2918:
2915:
2912:
2900:
2896:
2895:
2893:
2892:
2889:
2886:
2878:
2867:
2837:
2829:
2818:
2814:
2806:
2798:
2797:
2785:
2783:
2771:
2765:
2757:
2756:
2754:
2753:
2751:
2740:
2737:
2736:
2724:
2713:
2706:
2704:
2702:
2701:
2698:
2695:
2684:. Equivalently,
2683:
2681:
2679:
2678:
2675:
2672:
2660:
2653:
2650:For any integer
2647:
2645:
2644:
2639:
2634:
2618:
2617:
2602:
2599:
2595:
2591:
2590:
2585:
2571:
2536:
2533:
2487:
2486:
2484:
2483:
2480:
2477:
2465:
2462:of the function
2461:
2446:
2438:
2431:
2429:
2427:
2426:
2423:
2420:
2410:. Equivalently,
2409:
2399:
2392:
2386:
2384:
2383:
2378:
2373:
2357:
2356:
2341:
2338:
2334:
2330:
2329:
2324:
2313:
2260:
2257:
2167:
2161:
2150:
2143:
2139:
2127:
2121:
2113:
2106:
2105:
2103:
2102:
2099:
2096:
2084:
2077:
2070:
2066:
2062:
2058:
2056:
2055:
2050:
2011:
2007:
2006:
2003:
2000:
1994:
1993:
1990:
1987:
1981:
1979:
1978:
1973:
1971:
1970:
1948:
1947:
1944:
1941:
1935:
1934:
1931:
1928:
1914:
1907:
1887:
1867:
1847:
1818:
1816:
1815:
1810:
1808:
1807:
1798:
1773:
1770:
1766:
1761:
1747:
1738:
1713:
1710:
1706:
1698:
1658:
1654:
1587:
1585:
1584:
1581:
1578:
1571:
1569:
1568:
1565:
1562:
1530:PΓ³lya conjecture
1520:
1518:
1492:
1480:
1470:
1464:
1454:
1447:
1445:
1439:
1435:
1431:
1427:
1421:
1409:
1394:
1390:
1379:
1378:
1375:
1372:
1366:
1365:
1362:
1359:
1345:
1339:
1338:
1335:
1332:
1327:less than 10 is
1323:
1322:
1319:
1316:
1311:less than 10 is
1307:
1306:
1303:
1300:
1295:less than 10 is
1291:
1290:
1287:
1282:less than 10 is
1278:
1277:
1274:
1269:less than 10 is
1265:
1264:
1261:
1256:less than 10 is
1252:
1251:
1246:less than 10 is
1242:
1241:
1236:less than 10 is
1211:
1200:
1188:
1166:
1142:
878:
868:
858:
852:
837:
835:
834:
829:
827:
826:
817:
792:
789:
785:
780:
766:
757:
732:
729:
725:
717:
685:
681:
670:
663:
656:
652:
648:
640:
630:
626:
618:
602:
591:
589:
588:
583:
575:
574:
558:
556:
555:
550:
539:, there is some
538:
536:
535:
530:
512:
494:
490:
486:
483:is the value of
482:
473:
471:
470:
465:
463:
462:
447:
444:
437:
436:
399:
396:
378:
377:
356:
354:
353:
348:
346:
345:
336:
311:
308:
285:
260:
257:
250:
212:
189:positive integer
166:
162:
150:
146:
120:Jeffrey Lagarias
100:
98:
83:positive integer
35:
21:
18:Collatz sequence
9461:
9460:
9456:
9455:
9454:
9452:
9451:
9450:
9421:
9420:
9401:
9383:
9365:
9353:"Collatz Paths"
9350:
9341:Collatz Problem
9322:
9321:
9302:Wayback Machine
9284:Wayback Machine
9261:
9259:
9255:
9252:
9247:
9238:
9237:
9233:
9211:
9210:
9206:
9184:
9183:
9179:
9168:
9143:
9142:
9138:
9128:
9127:
9123:
9113:
9111:
9105:
9104:
9100:
9095:
9091:
9079:
9070:
9069:
9065:
9039:
9038:
9034:
9016:
9015:
9011:
8969:
8968:
8964:
8930:
8929:
8925:
8889:
8888:
8884:
8856:
8855:
8851:
8823:
8822:
8818:
8770:
8769:
8765:
8747:
8746:
8742:
8726:
8713:
8712:
8705:
8637:
8636:
8632:
8625:Quanta Magazine
8618:
8617:
8613:
8579:
8574:
8573:
8566:
8557:
8553:
8536:
8530:
8528:
8498:
8485:
8484:
8475:
8443:
8442:
8433:
8391:
8390:
8383:
8353:
8352:
8348:
8337:
8335:
8330:
8329:
8325:
8299:
8298:
8294:
8248:
8247:
8240:
8225:
8201:
8200:
8193:
8178:
8168:Springer-Verlag
8156:Guy, Richard K.
8154:
8153:
8149:
8142:
8120:
8119:
8115:
8108:
8087:
8086:
8082:
8072:
8071:
8067:
8058:
8054:
8018:
8017:
8000:
7990:
7977:
7976:
7972:
7968:
7963:
7962:
7932:Shizuo Kakutani
7920:Ulam conjecture
7905:
7890:
7887:
7883:
7878:
7848:
7838:
7831:
7828:
7808:
7774:
7769:
7768:
7764:
7757:
7753:
7748:
7745:
7744:
7743:
7741:
7725:
7721:
7710:
7706:
7692:halting problem
7683:
7672:
7668:
7664:
7656:
7650:
7646:
7640:
7636:
7630:
7622:
7619:
7616:
7615:
7613:
7611:
7605:
7598:
7587:
7584:
7574:
7564:
7557:
7551:
7494:
7478:
7457:
7456:
7442:
7436:
7420:
7413:
7409:
7405:
7394:
7391:
7384:
7383:
7381:
7372:
7368:
7355:
7340:
7336:
7329:
7310:
7291:
7281:
7264:
7259:
7247:
7243:
7239:
7228:
7223:
7219:
7214:
7205:
7197:
7193:
7190:
7178:
7174:
7170:
7166:
7162:
7154:
7153:, smaller than
7139:
7132:
7131:, smaller than
7118:
7114:
7110:
7106:
7069:
7062:
7058:
7054:
7050:
7037:
7030:
7022:
7016:
7006:
6991:
6987:
6983:
6979:
6964:
6960:
6956:
6952:
6948:
6944:
6909:
6902:
6898:
6894:
6890:
6886:
6882:
6873:
6868:
6806:
6805:
6768:
6767:
6731:
6724:
6720:
6659:
6658:
6617:
6616:
6574:
6573:
6531:
6530:
6527:critical points
6509:
6508:
6483:
6413:
6409:
6395:
6394:
6371:
6367:
6354:
6335:
6311:
6307:
6294:
6282:
6257:
6256:
6250:entire function
6221:
6220:
6184:
6167:
6166:
6095:
6091:
6074:
6073:
6043:
6042:
6007:
6006:
5974:
5973:
5954:
5953:
5930:
5929:
5877:
5873:
5860:
5838:
5833:
5832:
5800:
5796:
5783:
5761:
5756:
5755:
5721:
5699:
5673:
5640:
5639:
5616:
5615:
5605:
5596:
5595:
5585:
5572:
5549:
5544:
5543:
5525:
5524:
5514:
5505:
5504:
5494:
5481:
5458:
5453:
5452:
5428:
5423:
5422:
5401:
5396:
5395:
5372:
5371:
5329:
5328:
5300:
5299:
5280:
5279:
5252:
5251:
5233:
5179:
5175:
5167:
5166:
5140:
5135:
5134:
5122:
5096:
5063:
5062:
5058:
5017:
5016:
4993:
4988:
4987:
4983:
4965:2-adic integers
4941:
4936:
4935:
4918:
4917:
4885:
4866:
4861:
4860:
4828:
4811:
4790:
4789:
4786:
4765:
4764:
4729:
4710:
4705:
4704:
4669:
4652:
4629:
4624:
4623:
4612:
4608:
4601:
4597:
4587:
4584:
4581:
4580:
4578:
4574:
4570:
4524:
4511:
4510:
4499:
4489:
4476:
4466:
4453:
4443:
4430:
4420:
4419:
4412:
4411:
4408:(0 1 1 0 0 1 1)
4407:
4404:(1 0 1 1 0 0 1)
4403:
4281:
4280:
4243:
4230:
4229:
4218:
4208:
4195:
4185:
4172:
4162:
4149:
4139:
4138:
4131:
4130:
4127:(1 0 1 1 0 0 1)
4126:
4115:
4083:
4070:
4069:
4050:
4045:
4035:
4014:
4009:
3993:
3992:
3985:
3984:
3976:
3966:
3960:
3956:
3952:
3946:parity sequence
3939:2-adic integers
3934:
3925:
3909:
3903:
3897:
3891:
3886:β β164 β β82 β
3885:
3879:
3873:
3867:
3849:
3843:
3837:
3819:
3813:
3795:
3789:
3762:
3758:
3754:
3749:
3734:
3727:
3723:
3719:
3705:
3695:
3685:
3643:
3642:
3626:
3607:
3602:
3601:
3585:
3568:
3547:
3546:
3541:
3539:As a tag system
3532:
3528:
3520:
3512:
3508:
3500:
3492:
3488:
3484:
3480:
3472:
3468:
3464:
3460:
3452:
3444:
3436:
3428:
3424:
3420:
3416:
3412:
3408:
3397:
3383:
3379:
3375:
3368:
3364:
3360:
3356:
3352:
3348:
3337:
3326:
3323:
3318:
3317:
3315:
3314:
3306:
3303:
3296:
3295:
3293:
3292:
3285:
3272:
3263:
3256:
3250:
3245:
3238:
3233:
3229:
3221:
3213:
3209:
3171:
3170:
3154:
3135:
3130:
3129:
3113:
3096:
3075:
3074:
3071:
3066:
3065:
3007:
3004:
2994:
2975:
2964:
2960:
2952:
2937:
2928:
2924:
2916:
2913:
2910:
2909:
2907:
2906:
2898:
2890:
2887:
2884:
2883:
2881:
2880:
2877:
2869:
2866:
2859:
2850:
2842:
2835:
2820:
2816:
2812:
2804:
2795:
2791:
2786:is the highest
2781:
2773:
2763:
2759:
2749:
2741:
2738:
2734:
2730:
2729:
2727:
2726:
2718:
2708:
2707:if and only if
2699:
2696:
2689:
2688:
2686:
2685:
2676:
2673:
2666:
2665:
2663:
2662:
2661:if and only if
2655:
2651:
2613:
2612:
2596:
2572:
2560:
2556:
2553:
2552:
2530:
2511:
2490:
2489:
2481:
2478:
2471:
2470:
2468:
2467:
2463:
2455:
2447:defined in the
2444:
2433:
2432:if and only if
2424:
2421:
2415:
2414:
2412:
2411:
2408:+ 1 β‘ 4 (mod 6)
2403:
2394:
2390:
2352:
2351:
2335:
2314:
2302:
2298:
2295:
2294:
2254:
2235:
2214:
2213:
2180:
2175:
2163:
2156:
2145:
2141:
2137:
2123:
2119:
2116:
2111:
2100:
2097:
2094:
2093:
2091:
2090:
2079:
2072:
2068:
2064:
2060:
2014:
2013:
2009:
2004:
2001:
1998:
1996:
1991:
1988:
1985:
1983:
1962:
1951:
1950:
1945:
1942:
1939:
1937:
1932:
1929:
1926:
1924:
1921:
1912:
1906:
1898:
1889:
1886:
1879:
1869:
1866:
1859:
1849:
1844:
1838:
1831:
1824:
1803:
1802:
1767:
1748:
1743:
1742:
1707:
1690:
1669:
1668:
1665:
1656:
1650:
1643:
1634:Quanta Magazine
1602:
1582:
1579:
1576:
1575:
1573:
1566:
1563:
1560:
1559:
1557:
1550:
1526:counterexamples
1516:
1514:
1511:
1503:
1496:
1493:
1484:
1481:
1472:
1468:
1465:
1456:
1449:
1443:
1441:
1437:
1433:
1429:
1425:
1422:
1413:
1410:
1401:
1392:
1388:
1376:
1373:
1370:
1368:
1363:
1360:
1357:
1355:
1341:
1336:
1333:
1330:
1328:
1320:
1317:
1314:
1312:
1304:
1301:
1298:
1296:
1288:
1285:
1283:
1275:
1272:
1270:
1262:
1259:
1257:
1249:
1247:
1239:
1237:
1207:
1201:to reach 1 are
1198:
1184:
1162:
1138:
1133:
1128:, 16, 8, 4, 2,
1127:
1121:
1115:
1109:
1104:, 184, 92, 46,
1103:
1097:
1091:
1085:
1079:
1073:
1067:
1061:
1055:
1049:
1043:
1037:
1031:
1025:
1019:
1013:
1007:
1001:
995:
989:
983:
977:
971:
965:
959:
953:
947:
941:
935:
929:
923:
917:
911:
905:
899:
893:
887:
873:
863:
854:
847:
844:
822:
821:
786:
767:
762:
761:
726:
709:
688:
687:
683:
675:
665:
661:
654:
650:
646:
637:
632:
628:
624:
617:
609:
604:
600:
566:
561:
560:
541:
540:
521:
520:
501:
496:
492:
488:
484:
480:
475:
458:
457:
441:
422:
413:
412:
393:
383:
369:
364:
363:
341:
340:
305:
290:
289:
254:
236:
215:
214:
210:
164:
160:
148:
144:
129:
96:
94:
56:
55:
50:
48:
37:
33:
28:
23:
22:
15:
12:
11:
5:
9459:
9457:
9449:
9448:
9443:
9438:
9433:
9423:
9422:
9419:
9418:
9413:
9399:
9381:
9363:
9348:
9338:
9319:
9312:
9305:
9287:
9270:
9251:
9250:External links
9248:
9246:
9245:
9231:
9220:(5): 351β367.
9204:
9177:
9166:
9136:
9121:
9098:
9089:
9063:
9052:(3): 241β252.
9032:
9009:
8962:
8943:(2): 145β151.
8923:
8882:
8849:
8816:
8763:
8740:
8703:
8658:(3): 237β258.
8630:
8611:
8590:(3): 241β252.
8564:
8551:
8473:
8431:
8381:
8346:
8323:
8292:
8238:
8223:
8205:, ed. (2010).
8191:
8176:
8147:
8140:
8113:
8106:
8080:
8065:
8052:
7998:
7988:
7969:
7967:
7964:
7961:
7960:
7940:Bryan Thwaites
7924:StanisΕaw Ulam
7880:
7879:
7877:
7874:
7873:
7872:
7867:
7862:
7857:
7844:
7843:
7827:
7824:
7816:Turing machine
7807:
7804:
7781:
7777:
7763:Iterations of
7746:
7733:
7732:
7688:
7687:
7654:
7644:
7634:
7609:
7579:
7569:
7562:
7555:
7539:
7534:
7531:
7527:
7524:
7519:
7516:
7513:
7501:
7497:
7493:
7490:
7485:
7481:
7477:
7474:
7471:
7468:
7465:
7438:Main article:
7435:
7432:
7402:
7401:
7365:
7326:
7189:
7186:
7105:holds for all
7103:
7102:
7049:
7046:
7034:precomputation
7029:This requires
7027:
7026:
7020:
6943:The values of
6941:
6940:
6872:
6869:
6867:
6864:
6840:
6837:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6793:
6790:
6787:
6784:
6781:
6778:
6775:
6764:
6763:
6750:
6744:
6741:
6738:
6734:
6730:
6727:
6723:
6719:
6716:
6713:
6710:
6707:
6704:
6699:
6696:
6691:
6686:
6683:
6678:
6675:
6672:
6669:
6666:
6624:
6596:
6593:
6590:
6587:
6584:
6581:
6538:
6523:
6522:
6507:
6504:
6501:
6498:
6495:
6490:
6486:
6482:
6479:
6476:
6473:
6470:
6467:
6464:
6461:
6458:
6455:
6452:
6448:
6444:
6441:
6438:
6435:
6432:
6429:
6426:
6421:
6418:
6412:
6406:
6403:
6398:
6396:
6393:
6388:
6384:
6379:
6376:
6370:
6366:
6361:
6357:
6351:
6347:
6344:
6341:
6338:
6332:
6328:
6324:
6319:
6316:
6310:
6306:
6301:
6297:
6291:
6288:
6283:
6280:
6277:
6274:
6271:
6268:
6265:
6264:
6237:
6234:
6231:
6228:
6208:
6205:
6202:
6199:
6196:
6191:
6187:
6183:
6180:
6177:
6174:
6155:
6151:
6148:
6145:
6142:
6139:
6136:
6132:
6128:
6125:
6122:
6119:
6116:
6113:
6110:
6104:
6101:
6094:
6087:
6084:
6050:
6027:
6024:
6020:
6017:
6014:
5994:
5991:
5987:
5984:
5981:
5961:
5937:
5896:
5892:
5886:
5883:
5876:
5872:
5867:
5863:
5859:
5856:
5853:
5850:
5845:
5841:
5819:
5815:
5809:
5806:
5799:
5795:
5790:
5786:
5782:
5779:
5776:
5773:
5768:
5764:
5752:
5751:
5739:
5736:
5733:
5728:
5724:
5720:
5715:
5711:
5708:
5705:
5702:
5695:
5691:
5688:
5685:
5680:
5676:
5672:
5667:
5664:
5659:
5656:
5653:
5650:
5647:
5633:
5632:
5619:
5609:
5606:
5604:
5601:
5598:
5597:
5592: is even,
5589:
5586:
5584:
5581:
5578:
5577:
5575:
5570:
5567:
5564:
5561:
5556:
5552:
5541:
5528:
5518:
5515:
5513:
5510:
5507:
5506:
5501: is even,
5498:
5495:
5493:
5490:
5487:
5486:
5484:
5479:
5476:
5473:
5470:
5465:
5461:
5435:
5431:
5408:
5404:
5379:
5359:
5355:
5351:
5348:
5345:
5342:
5339:
5336:
5316:
5313:
5310:
5307:
5287:
5267:
5263:
5259:
5232:
5229:
5217:
5213:
5206:
5203:
5197:
5193:
5188:
5183:
5178:
5174:
5149:
5144:
5108:
5103:
5099:
5094:
5090:
5085:
5081:
5078:
5075:
5070:
5066:
5061:
5055:
5050:
5047:
5044:
5040:
5036:
5033:
5030:
5027:
5024:
5002:
4997:
4950:
4945:
4921:
4915:
4912:
4908:
4905:
4900:
4897:
4894:
4886:
4882:
4878:
4875:
4872:
4869:
4863:
4862:
4858:
4855:
4851:
4848:
4843:
4840:
4837:
4829:
4825:
4822:
4817:
4816:
4814:
4809:
4806:
4803:
4800:
4797:
4785:
4782:
4768:
4763:
4759:
4756:
4752:
4749:
4744:
4741:
4738:
4730:
4726:
4722:
4719:
4716:
4713:
4707:
4706:
4703:
4699:
4696:
4692:
4689:
4684:
4681:
4678:
4670:
4666:
4663:
4658:
4657:
4655:
4650:
4647:
4644:
4641:
4636:
4632:
4552:
4547:
4544:
4539:
4531:
4527:
4523:
4518:
4514:
4506:
4502:
4496:
4492:
4488:
4483:
4479:
4473:
4469:
4465:
4460:
4456:
4450:
4446:
4442:
4437:
4433:
4427:
4423:
4389:
4384:
4381:
4376:
4371:
4368:
4363:
4358:
4355:
4350:
4345:
4342:
4337:
4332:
4329:
4324:
4319:
4316:
4311:
4306:
4303:
4298:
4293:
4290:
4266:
4263:
4258:
4250:
4246:
4242:
4237:
4233:
4225:
4221:
4215:
4211:
4207:
4202:
4198:
4192:
4188:
4184:
4179:
4175:
4169:
4165:
4161:
4156:
4152:
4146:
4142:
4121:
4120:
4111:
4109:
4098:
4090:
4086:
4082:
4077:
4073:
4063:
4060:
4057:
4053:
4048:
4042:
4038:
4034:
4031:
4028:
4021:
4017:
4012:
4006:
4003:
4000:
3996:
3971:
3964:
3933:
3930:
3920:
3919:
3916:
3913:
3905:
3899:
3893:
3887:
3881:
3875:
3869:
3863:
3860:
3859:
3856:
3853:
3845:
3844:β β20 β β10 β
3839:
3833:
3830:
3829:
3826:
3823:
3815:
3809:
3806:
3805:
3802:
3799:
3791:
3785:
3782:
3781:
3778:
3775:
3753:
3750:
3748:
3745:
3716:
3715:
3663:
3660:
3656:
3653:
3646:
3641:
3638:
3635:
3627:
3623:
3619:
3616:
3613:
3610:
3604:
3603:
3600:
3597:
3594:
3586:
3582:
3579:
3574:
3573:
3571:
3566:
3563:
3560:
3557:
3554:
3540:
3537:
3281:
3267:
3254:
3243:
3236:
3195:
3191:
3188:
3184:
3181:
3174:
3169:
3166:
3163:
3155:
3151:
3147:
3144:
3141:
3138:
3132:
3131:
3128:
3125:
3122:
3114:
3110:
3107:
3102:
3101:
3099:
3094:
3091:
3088:
3085:
3082:
3070:
3067:
3013:
3012:
3003:
3000:
2999:
2998:
2991:
2972:
2936:
2933:
2873:
2864:
2854:
2846:
2637:
2633:
2630:
2626:
2623:
2616:
2611:
2608:
2605:
2597:
2594:
2588:
2584:
2581:
2578:
2575:
2569:
2566:
2563:
2559:
2555:
2554:
2551:
2548:
2545:
2542:
2539:
2531:
2529:
2526:
2523:
2520:
2517:
2516:
2514:
2509:
2506:
2503:
2500:
2497:
2401:if and only if
2376:
2372:
2369:
2365:
2362:
2355:
2350:
2347:
2344:
2336:
2333:
2327:
2323:
2320:
2317:
2311:
2308:
2305:
2301:
2297:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2255:
2253:
2250:
2247:
2244:
2241:
2240:
2238:
2233:
2230:
2227:
2224:
2221:
2179:
2176:
2174:
2171:
2115:
2109:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
1969:
1965:
1961:
1958:
1920:
1917:
1904:
1896:
1884:
1877:
1864:
1857:
1842:
1836:
1829:
1823:is a sequence
1806:
1801:
1797:
1794:
1790:
1787:
1782:
1779:
1776:
1768:
1764:
1760:
1757:
1754:
1751:
1745:
1744:
1741:
1737:
1734:
1730:
1727:
1722:
1719:
1716:
1708:
1704:
1701:
1696:
1695:
1693:
1688:
1685:
1682:
1679:
1676:
1664:
1661:
1642:
1639:
1610:parity vectors
1601:
1600:Stopping times
1598:
1549:
1546:
1510:
1507:
1502:
1499:
1498:
1497:
1494:
1487:
1485:
1482:
1475:
1473:
1466:
1459:
1457:
1423:
1416:
1414:
1411:
1404:
1400:
1399:Visualizations
1397:
1352:
1351:
1325:
1309:
1293:
1280:
1267:
1254:
1244:
1234:
1231:
1228:
1225:
1218:
1217:
1195:
1194:
1173:
1172:
1135:
1134:
1129:
1123:
1117:
1111:
1105:
1099:
1093:
1087:
1081:
1075:
1069:
1063:
1057:
1051:
1045:
1039:
1033:
1027:
1021:
1015:
1009:
1003:
997:
991:
985:
979:
973:
967:
961:
955:
949:
943:
937:
931:
925:
919:
913:
907:
901:
895:
889:
883:
843:
842:Empirical data
840:
825:
820:
816:
813:
809:
806:
801:
798:
795:
787:
783:
779:
776:
773:
770:
764:
763:
760:
756:
753:
749:
746:
741:
738:
735:
727:
723:
720:
715:
714:
712:
707:
704:
701:
698:
695:
641:is called the
635:
619:is called the
615:
607:
581:
578:
573:
569:
548:
528:
499:
478:
461:
456:
453:
450:
442:
440:
435:
432:
429:
425:
421:
418:
415:
414:
411:
408:
405:
402:
394:
392:
389:
388:
386:
381:
376:
372:
344:
339:
335:
332:
328:
325:
320:
317:
314:
306:
304:
301:
298:
295:
292:
291:
288:
284:
281:
277:
274:
269:
266:
263:
255:
253:
249:
245:
242:
241:
239:
234:
231:
228:
225:
222:
200:
199:
196:
128:
125:
106:Lothar Collatz
63:Directed graph
51:
47:
46:
43:
39:
38:
32:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
9458:
9447:
9444:
9442:
9439:
9437:
9434:
9432:
9429:
9428:
9426:
9417:
9414:
9410:
9409:
9404:
9400:
9396:
9392:
9391:
9386:
9382:
9378:
9374:
9373:
9368:
9364:
9360:
9359:
9354:
9349:
9346:
9342:
9339:
9334:
9333:
9328:
9325:
9320:
9317:
9313:
9310:
9306:
9303:
9299:
9296:
9292:
9288:
9285:
9281:
9278:
9275:
9271:
9267:
9254:
9253:
9249:
9241:
9235:
9232:
9227:
9223:
9219:
9215:
9208:
9205:
9200:
9196:
9192:
9188:
9187:Computability
9181:
9178:
9175:
9169:
9163:
9159:
9155:
9151:
9147:
9140:
9137:
9132:
9125:
9122:
9110:. p. 208
9109:
9102:
9099:
9093:
9090:
9085:
9078:
9076:
9067:
9064:
9059:
9055:
9051:
9047:
9043:
9036:
9033:
9029:(4): 495β509.
9028:
9024:
9020:
9013:
9010:
9005:
9001:
8996:
8991:
8987:
8983:
8982:
8977:
8975:
8966:
8963:
8958:
8954:
8950:
8946:
8942:
8938:
8934:
8927:
8924:
8919:
8915:
8910:
8905:
8901:
8897:
8893:
8886:
8883:
8877:
8872:
8868:
8864:
8860:
8853:
8850:
8844:
8839:
8835:
8831:
8827:
8820:
8817:
8812:
8808:
8803:
8798:
8794:
8790:
8786:
8782:
8778:
8776:
8767:
8764:
8759:
8755:
8751:
8744:
8741:
8736:
8732:
8725:
8723:
8720:-cycles with
8719:
8710:
8708:
8704:
8699:
8695:
8691:
8687:
8683:
8679:
8675:
8671:
8666:
8661:
8657:
8653:
8649:
8647:
8641:
8634:
8631:
8626:
8622:
8615:
8612:
8607:
8603:
8598:
8593:
8589:
8585:
8578:
8571:
8569:
8565:
8561:
8555:
8552:
8547:
8541:
8525:
8520:
8516:
8512:
8508:
8504:
8497:
8495:
8491:
8482:
8480:
8478:
8474:
8468:
8463:
8459:
8455:
8451:
8449:
8440:
8438:
8436:
8432:
8427:
8423:
8418:
8413:
8409:
8405:
8404:
8399:
8397:
8388:
8386:
8382:
8377:
8373:
8369:
8365:
8361:
8357:
8350:
8347:
8334:
8327:
8324:
8319:
8315:
8312:(11): 79β99.
8311:
8307:
8303:
8296:
8293:
8288:
8284:
8279:
8274:
8269:
8264:
8260:
8256:
8252:
8245:
8243:
8239:
8234:
8230:
8226:
8220:
8216:
8212:
8208:
8204:
8198:
8196:
8192:
8187:
8183:
8179:
8177:0-387-20860-7
8173:
8169:
8165:
8161:
8157:
8151:
8148:
8143:
8141:0-465-02685-0
8137:
8133:
8129:
8128:
8123:
8117:
8114:
8109:
8107:0-19-513342-0
8103:
8099:
8094:
8093:
8084:
8081:
8076:
8069:
8066:
8062:
8056:
8053:
8048:
8044:
8040:
8036:
8032:
8028:
8027:
8022:
8015:
8013:
8011:
8009:
8007:
8005:
8003:
7999:
7995:
7991:
7989:0-7890-0374-0
7985:
7981:
7974:
7971:
7965:
7957:
7953:
7949:
7945:
7941:
7937:
7933:
7929:
7925:
7921:
7917:
7913:
7909:
7902:
7898:
7894:
7885:
7882:
7875:
7871:
7868:
7866:
7863:
7861:
7858:
7856:
7852:
7846:
7845:
7841:
7835:
7830:
7825:
7823:
7821:
7817:
7813:
7805:
7803:
7801:
7797:
7779:
7775:
7761:
7739:
7728:
7717:
7713:
7704:
7703:
7702:
7700:
7695:
7694:in this way.
7693:
7679:
7675:
7662:
7661:
7660:
7653:
7643:
7633:
7608:
7601:
7594:
7590:
7582:
7578:
7572:
7568:
7561:
7554:
7537:
7529:
7525:
7517:
7514:
7511:
7499:
7495:
7491:
7488:
7483:
7479:
7475:
7469:
7463:
7453:
7451:
7447:
7441:
7433:
7431:
7427:
7423:
7416:
7388:
7379:
7375:
7366:
7363:
7358:
7351:
7347:
7343:
7332:
7327:
7322:
7318:
7315:+ 1) + 1 = 12
7314:
7306:
7302:
7298:
7294:
7288:
7284:
7279:
7278:
7277:
7274:
7272:
7267:
7258:
7254:
7250:
7242:from the set
7237:
7231:
7222:
7213:
7209:
7201:
7187:
7185:
7182:
7158:
7150:
7146:
7142:
7136:
7129:
7125:
7121:
7100:
7096:
7092:
7088:
7084:
7080:
7076:
7072:
7068:
7067:
7066:
7047:
7045:
7043:
7035:
7021:
7015:
7014:
7013:
7009:
7002:
6998:
6994:
6975:
6971:
6967:
6959:-bit numbers
6936:
6932:
6928:
6924:
6920:
6916:
6912:
6908:
6907:
6906:
6880:
6879:
6870:
6866:Optimizations
6865:
6863:
6861:
6857:
6838:
6835:
6829:
6826:
6823:
6817:
6811:
6791:
6788:
6782:
6776:
6773:
6748:
6742:
6739:
6736:
6732:
6728:
6725:
6721:
6714:
6711:
6708:
6705:
6697:
6694:
6689:
6684:
6681:
6676:
6670:
6664:
6657:
6656:
6655:
6653:
6644:
6640:
6638:
6622:
6614:
6610:
6594:
6591:
6585:
6579:
6567:
6562:
6558:
6556:
6552:
6536:
6528:
6502:
6499:
6493:
6488:
6484:
6477:
6471:
6468:
6462:
6459:
6453:
6450:
6446:
6439:
6436:
6430:
6427:
6424:
6419:
6416:
6410:
6404:
6401:
6391:
6386:
6382:
6377:
6374:
6368:
6364:
6359:
6355:
6349:
6345:
6342:
6339:
6336:
6330:
6326:
6322:
6317:
6314:
6308:
6304:
6299:
6295:
6289:
6286:
6278:
6272:
6266:
6255:
6254:
6253:
6251:
6232:
6226:
6203:
6200:
6194:
6189:
6185:
6178:
6172:
6153:
6146:
6143:
6137:
6134:
6130:
6123:
6120:
6114:
6111:
6108:
6102:
6099:
6092:
6085:
6082:
6071:
6067:
6066:complex plane
6062:
6048:
6041:
6022:
6018:
6015:
5989:
5985:
5982:
5959:
5951:
5935:
5927:
5926:monotonically
5923:
5920:, as well as
5919:
5915:
5911:
5894:
5890:
5884:
5881:
5874:
5870:
5865:
5861:
5857:
5851:
5843:
5839:
5817:
5813:
5807:
5804:
5797:
5793:
5788:
5784:
5780:
5774:
5766:
5762:
5734:
5726:
5722:
5718:
5713:
5709:
5706:
5703:
5700:
5693:
5686:
5678:
5674:
5670:
5665:
5662:
5657:
5651:
5645:
5638:
5637:
5636:
5612: is odd,
5607:
5602:
5599:
5587:
5582:
5579:
5573:
5568:
5562:
5554:
5550:
5542:
5521: is odd,
5516:
5511:
5508:
5496:
5491:
5488:
5482:
5477:
5471:
5463:
5459:
5451:
5450:
5449:
5433:
5429:
5406:
5402:
5393:
5392:interpolating
5377:
5357:
5353:
5346:
5343:
5340:
5337:
5314:
5311:
5308:
5305:
5285:
5265:
5261:
5257:
5249:
5241:
5237:
5230:
5228:
5215:
5204:
5201:
5195:
5191:
5186:
5176:
5172:
5163:
5147:
5132:
5128:
5121:The function
5119:
5106:
5101:
5097:
5092:
5088:
5083:
5076:
5068:
5064:
5059:
5048:
5045:
5042:
5038:
5034:
5028:
5022:
5000:
4981:
4980:parity vector
4976:
4974:
4970:
4966:
4948:
4910:
4906:
4898:
4895:
4892:
4880:
4876:
4873:
4870:
4867:
4853:
4849:
4841:
4838:
4835:
4823:
4820:
4812:
4807:
4801:
4795:
4788:The function
4783:
4781:
4761:
4754:
4750:
4742:
4739:
4736:
4724:
4720:
4717:
4714:
4711:
4701:
4694:
4690:
4682:
4679:
4676:
4664:
4661:
4653:
4648:
4642:
4634:
4630:
4620:
4616:
4605:
4594:
4568:
4563:
4550:
4545:
4542:
4537:
4529:
4525:
4521:
4516:
4512:
4504:
4500:
4494:
4490:
4486:
4481:
4477:
4471:
4467:
4463:
4458:
4454:
4448:
4444:
4440:
4435:
4431:
4425:
4421:
4400:
4387:
4382:
4379:
4369:
4366:
4356:
4353:
4343:
4340:
4330:
4327:
4317:
4314:
4304:
4301:
4291:
4288:
4264:
4261:
4256:
4248:
4244:
4240:
4235:
4231:
4223:
4219:
4213:
4209:
4205:
4200:
4196:
4190:
4186:
4182:
4177:
4173:
4167:
4163:
4159:
4154:
4150:
4144:
4140:
4119:
4112:
4110:
4096:
4088:
4084:
4080:
4075:
4071:
4061:
4058:
4055:
4051:
4046:
4040:
4036:
4032:
4029:
4026:
4019:
4015:
4010:
4004:
4001:
3998:
3994:
3983:
3982:
3979:
3974:
3970:
3963:
3949:
3947:
3942:
3940:
3931:
3929:
3917:
3914:
3912:
3908:
3902:
3896:
3890:
3884:
3878:
3872:
3866:
3862:
3861:
3857:
3854:
3852:
3848:
3842:
3836:
3832:
3831:
3827:
3824:
3822:
3818:
3812:
3808:
3807:
3803:
3800:
3798:
3794:
3788:
3784:
3783:
3779:
3776:
3773:
3772:
3769:
3766:
3751:
3746:
3744:
3742:
3741:
3731:
3712:
3708:
3702:
3698:
3692:
3688:
3684:
3683:
3682:
3680:
3675:
3658:
3654:
3639:
3636:
3633:
3621:
3617:
3614:
3611:
3608:
3598:
3595:
3592:
3580:
3577:
3569:
3564:
3558:
3552:
3544:
3538:
3536:
3524:
3516:
3504:
3496:
3476:
3456:
3448:
3440:
3432:
3405:
3401:
3394:
3390:
3386:
3374:Applying the
3372:
3344:
3340:
3334:
3321:
3300:
3289:
3284:
3280:
3276:
3270:
3266:
3260:
3253:
3246:
3239:
3225:
3217:
3206:
3193:
3186:
3182:
3167:
3164:
3161:
3149:
3145:
3142:
3139:
3136:
3126:
3123:
3120:
3108:
3105:
3097:
3092:
3086:
3080:
3068:
3064:
3060:
3059:
3054:
3050:
3049:
3044:
3040:
3039:
3034:
3030:
3029:
3024:
3020:
3019:
3011:
3001:
2992:
2987:
2983:
2979:
2973:
2968:
2958:
2957:
2956:
2950:
2946:
2943:that handles
2942:
2934:
2932:
2904:
2876:
2872:
2863:
2857:
2853:
2849:
2845:
2841:
2834:, the number
2833:
2827:
2823:
2810:
2802:
2794:
2790:that divides
2789:
2780:
2776:
2769:
2762:
2748:
2744:
2733:
2722:
2715:
2711:
2693:
2670:
2658:
2648:
2635:
2628:
2624:
2609:
2606:
2603:
2592:
2586:
2582:
2579:
2576:
2573:
2567:
2564:
2561:
2557:
2549:
2546:
2543:
2540:
2537:
2524:
2521:
2512:
2507:
2501:
2495:
2475:
2459:
2452:
2450:
2442:
2436:
2418:
2407:
2402:
2397:
2387:
2374:
2367:
2363:
2348:
2345:
2342:
2331:
2325:
2321:
2318:
2315:
2309:
2306:
2303:
2299:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2248:
2245:
2236:
2231:
2225:
2219:
2212:
2208:
2204:
2203:Collatz graph
2200:
2199:Collatz graph
2192:
2191:
2184:
2177:
2172:
2170:
2166:
2159:
2154:
2148:
2144:-cycle up to
2134:
2132:
2126:
2110:
2108:
2089:expansion of
2088:
2082:
2075:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
1967:
1963:
1959:
1956:
1918:
1916:
1909:
1903:
1899:
1892:
1883:
1876:
1872:
1863:
1856:
1852:
1845:
1835:
1828:
1822:
1799:
1792:
1788:
1780:
1777:
1774:
1762:
1758:
1755:
1752:
1749:
1739:
1732:
1728:
1720:
1717:
1714:
1702:
1699:
1691:
1686:
1680:
1674:
1662:
1660:
1653:
1648:
1640:
1638:
1636:
1635:
1630:
1626:
1622:
1617:
1615:
1612:and uses the
1611:
1607:
1604:As proven by
1599:
1597:
1595:
1591:
1555:
1547:
1545:
1542:
1537:
1535:
1531:
1527:
1522:
1508:
1506:
1500:
1491:
1486:
1479:
1474:
1463:
1458:
1452:
1420:
1415:
1408:
1403:
1398:
1396:
1386:
1385:powers of two
1381:
1349:
1344:
1326:
1310:
1294:
1281:
1268:
1255:
1245:
1235:
1232:
1229:
1226:
1223:
1222:
1221:
1215:
1210:
1204:
1203:
1202:
1192:
1187:
1182:
1181:
1180:
1178:
1170:
1165:
1160:
1159:
1158:
1152:
1148:
1146:
1141:
1132:
1126:
1120:
1114:
1108:
1102:
1096:
1092:, 1300, 650,
1090:
1086:, 1732, 866,
1084:
1078:
1072:
1066:
1060:
1054:
1048:
1042:
1036:
1030:
1024:
1020:, 1276, 638,
1018:
1012:
1008:, 1132, 566,
1006:
1000:
994:
988:
984:, 1780, 890,
982:
976:
970:
964:
958:
952:
946:
940:
934:
928:
922:
916:
910:
904:
898:
892:
886:
882:
881:
880:
876:
870:
866:
860:
857:
850:
841:
839:
818:
811:
807:
799:
796:
793:
781:
777:
774:
771:
768:
758:
751:
747:
739:
736:
733:
721:
718:
710:
705:
699:
693:
679:
672:
668:
658:
644:
638:
622:
621:stopping time
614:
610:
599:The smallest
597:
593:
579:
576:
571:
567:
546:
526:
519:
514:
510:
506:
502:
481:
454:
451:
448:
433:
430:
427:
423:
416:
409:
406:
403:
400:
390:
384:
379:
374:
370:
362:In notation:
360:
357:
337:
330:
326:
318:
315:
312:
302:
299:
296:
293:
286:
279:
275:
267:
264:
261:
251:
247:
243:
237:
232:
226:
220:
209:
205:
197:
194:
193:
192:
190:
181:
173:
157:
141:
133:
126:
124:
121:
117:
113:
111:
107:
102:
92:
88:
84:
80:
76:
68:
64:
60:
54:
44:
41:
40:
30:
19:
9407:
9389:
9385:Eisenbud, D.
9371:
9367:Eisenbud, D.
9356:
9330:
9234:
9217:
9213:
9207:
9193:(1): 19β56.
9190:
9186:
9180:
9149:
9139:
9130:
9124:
9112:. Retrieved
9101:
9092:
9083:
9074:
9066:
9049:
9045:
9041:
9035:
9026:
9022:
9018:
9012:
8985:
8979:
8973:
8965:
8940:
8936:
8926:
8902:(1): 33β53.
8899:
8895:
8885:
8866:
8862:
8852:
8833:
8829:
8819:
8784:
8780:
8777:+ 1 problem"
8774:
8766:
8749:
8743:
8734:
8730:
8721:
8717:
8665:math/0205002
8655:
8651:
8645:
8633:
8624:
8614:
8587:
8583:
8554:
8540:cite journal
8529:. Retrieved
8509:(1): 51β70.
8506:
8502:
8493:
8489:
8460:(1): 45β56.
8457:
8453:
8447:
8410:(1): 19β22.
8407:
8401:
8395:
8359:
8355:
8349:
8336:. Retrieved
8326:
8309:
8305:
8301:
8295:
8258:
8254:
8210:
8206:
8163:
8150:
8126:
8116:
8091:
8083:
8068:
8055:
8030:
8024:
8020:
7993:
7979:
7973:
7951:
7948:Helmut Hasse
7943:
7935:
7927:
7919:
7915:
7907:
7904:
7900:
7892:
7889:
7884:
7850:
7809:
7762:
7734:
7726:
7715:
7711:
7698:
7696:
7689:
7677:
7673:
7651:
7641:
7631:
7606:
7599:
7592:
7588:
7580:
7576:
7570:
7566:
7559:
7552:
7454:
7443:
7425:
7421:
7414:
7403:
7386:
7377:
7373:
7367:For all odd
7356:
7349:
7348:β 1) = 2 Γ 3
7345:
7341:
7330:
7320:
7316:
7312:
7304:
7300:
7296:
7292:
7286:
7282:
7275:
7256:
7252:
7248:
7235:
7229:
7220:
7211:
7207:
7204:is even, so
7199:
7191:
7183:
7156:
7148:
7144:
7140:
7134:
7127:
7123:
7119:
7104:
7098:
7094:
7090:
7086:
7082:
7078:
7074:
7070:
7051:
7028:
7007:
7000:
6996:
6992:
6973:
6969:
6965:
6942:
6934:
6930:
6926:
6922:
6918:
6914:
6910:
6905:is given by
6876:
6875:The section
6874:
6859:
6855:
6765:
6649:
6571:
6551:Baker domain
6524:
6063:
5918:fixed points
5753:
5634:
5245:
5164:
5125:is a 2-adic
5120:
4979:
4977:
4787:
4618:
4614:
4606:
4595:
4566:
4564:
4401:
4124:
4113:
3972:
3968:
3967:< β― <
3961:
3950:
3943:
3935:
3923:
3910:
3906:
3900:
3894:
3888:
3882:
3876:
3870:
3864:
3850:
3846:
3840:
3834:
3820:
3816:
3810:
3796:
3792:
3786:
3767:
3755:
3738:
3732:
3717:
3710:
3706:
3700:
3696:
3690:
3686:
3679:2-tag system
3676:
3545:
3542:
3522:
3514:
3502:
3494:
3474:
3454:
3446:
3438:
3430:
3403:
3399:
3392:
3388:
3384:
3373:
3342:
3338:
3335:
3319:
3298:
3290:
3282:
3278:
3274:
3268:
3264:
3258:
3251:
3241:
3234:
3223:
3215:
3207:
3072:
3062:
3057:
3055:
3052:
3047:
3045:
3042:
3037:
3035:
3032:
3027:
3025:
3022:
3017:
3015:
3014:111
3005:
2985:
2981:
2977:
2966:
2938:
2874:
2870:
2861:
2855:
2851:
2847:
2843:
2830:). Then in
2825:
2821:
2792:
2778:
2774:
2767:
2760:
2746:
2742:
2731:
2720:
2716:
2709:
2691:
2668:
2656:
2649:
2473:
2457:
2453:
2434:
2416:
2405:
2395:
2388:
2202:
2198:
2196:
2187:
2164:
2157:
2155:-cycle with
2152:
2146:
2135:
2130:
2124:
2117:
2080:
2073:
1922:
1919:Cycle length
1910:
1901:
1894:
1890:
1881:
1874:
1870:
1861:
1854:
1850:
1840:
1833:
1826:
1666:
1651:
1644:
1641:Lower bounds
1632:
1618:
1603:
1553:
1551:
1541:lower bounds
1538:
1523:
1512:
1504:
1450:
1382:
1353:
1219:
1196:
1174:
1156:
1136:
1130:
1124:
1118:
1112:
1106:
1100:
1094:
1088:
1082:
1076:
1070:
1064:
1058:
1052:
1046:
1040:
1034:
1028:
1022:
1016:
1010:
1004:
998:
992:
986:
980:
974:
968:
962:
960:, 700, 350,
956:
950:
944:
942:, 412, 206,
938:
932:
930:, 364, 182,
926:
924:, 484, 242,
920:
914:
908:
902:
896:
890:
884:
874:
871:
864:
861:
855:
848:
845:
677:
673:
666:
659:
642:
633:
620:
612:
605:
598:
594:
517:
515:
508:
504:
497:
491:recursively
476:
361:
358:
213:as follows:
201:
186:
114:
103:
74:
72:
65:showing the
29:
9431:Conjectures
9272:An ongoing
8869:: 135β141.
8787:: 1565β72.
8211:+ 1 Problem
8033:(1): 3β23.
7812:Busy Beaver
7450:undecidable
7309:. (Because
6947:(or better
5240:Cobweb plot
4978:Define the
4567:irreducible
2868:where each
2809:odd numbers
2712:β‘ 2 (mod 3)
2705:β‘ 1 (mod 2)
2682:β‘ 2 (mod 3)
2659:β‘ 1 (mod 2)
2437:β‘ 4 (mod 6)
2430:β‘ 1 (mod 2)
2398:β‘ 1 (mod 2)
1888:, ..., and
1621:Terence Tao
1606:Riho Terras
1436:axis: some
894:, 124, 62,
862:The number
487:applied to
9425:Categories
9345:PlanetMath
8781:Math. Comp
8722:m <= 91
8531:2023-03-28
8268:1909.03562
8233:1253.11003
8186:1058.11001
7966:References
7950:), or the
7916:conjecture
7901:conjecture
7820:Paul ErdΕs
7724:, for all
7419:such that
7263:(sequence
6564:A Collatz
5950:attracting
5910:iterations
5131:almost all
4986:acting on
3790:β 4 β 2 β
3740:Tag system
3726:copies of
2807:maps from
2788:power of 2
2178:In reverse
1629:almost all
1594:almost all
1391:is halved
1137:(sequence
631:such that
603:such that
474:(that is:
116:Paul ErdΕs
110:hailstones
9332:MathWorld
9004:0008-414X
8918:0065-1036
8376:220294340
8287:2050-5086
7855:semigroup
7760:to 6480.
7701:problem:
7515:≡
7444:In 1972,
7319:+ 4 = 4(3
6986:times to
6982:function
6824:≈
6789:≫
6777:
6740:π
6729:−
6677:≜
6613:Julia set
6592:≜
6500:π
6494:
6460:π
6454:
6437:π
6431:
6425:−
6405:π
6375:π
6365:
6315:π
6305:
6279:≜
6201:π
6195:
6144:π
6138:
6121:π
6115:
6109:−
6086:π
6023:2.1386...
6016:1.1925...
5924:escaping
5882:π
5871:
5858:≜
5805:π
5794:
5781:≜
5719:⋅
5671:⋅
5658:≜
5448:, where:
5248:real line
5196:⊂
5054:∞
5039:∑
4982:function
4896:≡
4839:≡
4740:≡
4680:≡
4575:(1 1 0 0)
4522:−
4375:→
4362:→
4349:→
4336:→
4323:→
4310:→
4297:→
4241:−
4081:−
4059:−
4030:⋯
4002:−
3898:β β182 β
3892:β β122 β
3880:β β110 β
3637:≡
3596:≡
3419:times to
3415:function
3378:function
3165:≡
3124:≡
2955:remains:
2903:repetends
2801:remainder
2799:(with no
2607:≡
2580:−
2541:≡
2346:≡
2319:−
2265:≡
1960:×
1778:≡
1718:≡
1619:In 2019,
797:≡
737:≡
445:for
431:−
397:for
316:≡
265:≡
9395:Archived
9377:Archived
9298:Archived
9280:Archived
8957:17925995
8698:18467460
8642:(2003).
8338:14 March
8158:(2004).
8124:(1979).
7826:See also
7800:FRACTRAN
7440:FRACTRAN
7354:. (Here
7335:and odd
7280:For all
7227:odd and
7147:+ 1) = 3
7093:) < 2
7053:of
6963:, where
6219:, where
5948:has two
5127:isometry
4889:if
4832:if
4733:if
4673:if
3874:β β74 β
3868:β β50 β
3838:β β14 β
3630:if
3589:if
3519:becomes
3499:becomes
3407:, where
3249:, where
3226:+ 1) = 1
3158:if
3117:if
2600:if
2534:if
2339:if
2258:if
2211:relation
2188:Collatz
2044:85137581
2035:17087915
1771:if
1711:if
1074:, 6154,
1068:, 4102,
1062:, 2734,
1050:, 4858,
1044:, 3238,
1038:, 2158,
1032:, 1438,
978:, 1186,
790:if
730:if
507: (
495:times;
309:if
258:if
208:function
9316:project
9309:project
9295:project
9277:project
9114:26 July
8811:2137019
8789:Bibcode
8758:0535032
8690:1980260
8670:Bibcode
8606:0568274
8511:Bibcode
8426:2044308
8261:: e12.
8047:2322189
7954:(after
7946:(after
7938:(after
7934:), the
7930:(after
7922:(after
7918:), the
7912:problem
7903:), the
7897:problem
7798:called
7626:
7614:
7565:, ...,
7398:
7382:
7380:β 1) β€
7299:+ 1) =
7269:in the
7266:A075677
7234:. The
7210:+ 1 = 2
7113:modulo
6804:, then
6637:fractal
6566:fractal
6248:is any
6040:measure
4973:ergodic
4591:
4579:
3814:β β2 β
3477:+ 3 β 1
3459:. When
3330:
3316:
3310:
3294:
3002:Example
2959:Append
2945:strings
2920:
2908:
2894:
2882:
2840:strings
2755:
2728:
2703:
2687:
2680:
2664:
2485:
2469:
2428:
2413:
2131:1-cycle
2114:-cycles
2104:
2092:
1839:, ...,
1627:, that
1625:density
1586:
1574:
1570:
1558:
1346:in the
1343:A284668
1212:in the
1209:A006577
1189:in the
1186:A006884
1177:maximum
1167:in the
1164:A006877
1143:in the
1140:A008884
1116:, 106,
1026:, 958,
1014:, 850,
1002:, 754,
996:, 502,
972:, 790,
966:, 526,
954:, 466,
948:, 310,
936:, 274,
918:, 322,
912:, 214,
906:, 142,
9164:
9002:
8972:"The 3
8955:
8916:
8809:
8756:
8696:
8688:
8604:
8446:"The 3
8424:
8374:
8285:
8231:
8221:
8184:
8174:
8138:
8104:
8100:β118.
8045:
7986:
7729:> 0
7720:reach
7705:Given
7682:reach
7663:Given
7550:where
6990:, and
6951:) and
5922:orbits
5908:. The
3443:; for
3423:, and
3262:, and
3210:P(...)
2980:+ 1 +
2832:binary
2758:where
2201:. The
2138:(1; 2)
2059:where
2026:301994
1663:Cycles
1590:2-adic
1453:= 9663
1387:since
1110:, 70,
900:, 94,
888:, 82,
674:Since
67:orbits
9291:BOINC
9266:page"
9080:(PDF)
8953:S2CID
8727:(PDF)
8694:S2CID
8660:arXiv
8580:(PDF)
8499:(PDF)
8422:JSTOR
8372:S2CID
8263:arXiv
8132:400β2
8043:JSTOR
7876:Notes
7428:) = 1
7218:with
7081:) = 3
6921:) = 3
6893:(the
6858:, or
6856:hairs
5278:when
4602:(0 1)
4598:(1 0)
3774:Cycle
3465:2 β 1
3218:) = 0
3036:10001
3031:10001
2828:) = 1
2725:with
2207:graph
2205:is a
2190:graph
1913:(1,2)
1821:cycle
1645:In a
1448:(for
611:<
559:with
167:axis.
151:axis.
9162:ISBN
9116:2024
9000:ISSN
8914:ISSN
8546:link
8340:2020
8283:ISSN
8219:ISBN
8172:ISBN
8136:ISBN
8102:ISBN
7984:ISBN
7914:(or
7899:(or
7736:the
7667:and
7323:+ 1)
7271:OEIS
7255:) =
7126:) =
7061:and
7040:, a
6860:rays
6005:and
5831:and
5421:and
4600:and
3511:and
3363:and
3351:and
3240:= P(
3220:and
3063:0000
3046:1101
3041:1101
3026:1011
3021:1011
2949:bits
2860:...
2772:and
2441:tree
2160:β€ 91
2149:= 68
2101:ln 2
2095:ln 3
2078:and
2067:and
1936:(or
1900:) =
1880:) =
1860:) =
1532:and
1515:2.95
1424:The
1348:OEIS
1214:OEIS
1191:OEIS
1169:OEIS
1145:OEIS
1077:3077
1071:2051
1065:1367
1053:2429
1047:1619
1041:1079
877:= 27
867:= 19
851:= 12
452:>
95:2.95
91:even
73:The
9343:at
9264:+ 1
9222:doi
9195:doi
9174:PDF
9172:As
9154:doi
9054:doi
8990:doi
8945:doi
8904:doi
8871:doi
8867:618
8838:doi
8834:412
8797:doi
8678:doi
8656:109
8592:doi
8519:doi
8507:117
8462:doi
8458:118
8412:doi
8364:doi
8314:doi
8273:doi
8229:Zbl
8182:Zbl
8098:116
8035:doi
7942:),
7926:),
7910:+ 1
7895:+ 1
7853:+ 1
7657:= 1
7647:= 3
7637:= 0
7602:= 2
7583:β 1
7573:β 1
7526:mod
7417:β₯ 1
7408:in
7389:β 1
7360:is
7352:β 1
7333:β₯ 1
7311:3(4
7273:).
7232:β₯ 1
7202:+ 1
7192:If
7159:+ 1
7151:+ 1
7010:= 5
6615:of
6529:of
6485:sin
6451:sin
6428:cos
6356:sin
6296:cos
6186:sin
6135:sin
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694:f
684:n
678:n
676:3
667:n
662:n
655:k
651:i
647:n
636:k
634:a
629:k
625:n
616:0
613:a
608:i
606:a
601:i
580:1
577:=
572:i
568:a
547:i
527:n
511:)
509:n
505:f
500:i
498:a
493:i
489:n
485:f
479:i
477:a
455:0
449:i
439:)
434:1
428:i
424:a
420:(
417:f
410:,
407:0
404:=
401:i
391:n
385:{
380:=
375:i
371:a
338:.
334:)
331:2
324:(
319:1
313:n
303:1
300:+
297:n
294:3
287:,
283:)
280:2
273:(
268:0
262:n
252:2
248:/
244:n
238:{
233:=
230:)
227:n
224:(
221:f
211:f
165:y
161:x
149:y
145:x
97:Γ
36::
20:)
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