1282:
4660:
2460:
3240:
4629:
40:
3049:
2319:
3138:
problems involve an input sequence of items with fractional sizes, which must be placed into bins whose capacity (the total size of items placed into each bin) is one. Research into these problems has included the study of restricted bin packing problems where the item sizes are unit fractions.
3036:, unit fractions are often introduced earlier than other kinds of fractions, because of the ease of explaining them visually as equal parts of a whole. A common practical use of unit fractions is to divide food equally among a number of people, and exercises in performing this sort of
78:
Multiplying two unit fractions produces another unit fraction, but other arithmetic operations do not preserve unit fractions. In modular arithmetic, unit fractions can be converted into equivalent whole numbers, allowing modular division to be transformed into multiplication. Every
1249:
1757:
2188:
1262:. There is still interest today in analyzing the methods used by the ancients to choose among the possible representations for a fractional number, and to calculate with such representations. The topic of Egyptian fractions has also seen interest in modern
3146:, in which a collection of messages of equal length must each be repeatedly broadcast on a limited number of communication channels, with each message having a maximum delay between the start times of its repeated broadcasts. An item whose delay is
1857:
2686:
1951:
3285:
was a unit fraction. He initially thought it to be 1/136 and later changed his theory to 1/137. This contention has been falsified, given that current estimates of the fine structure constant are (to 6 significant digits) 1/137.036.
2050:
3226:
Even for bin packing problems with arbitrary item sizes, it can be helpful to round each item size up to the next larger unit fraction, and then apply a bin packing algorithm specialized for unit fraction sizes. In particular, the
1577:
1614:
1467:
314:
1133:
516:
3194:
of the time slots on the channel it is assigned to, so a solution to the scheduling problem can only come from a solution to the unit fraction bin packing problem with the channels as bins and the fractions
2850:
459:
392:
931:
2314:{\displaystyle {\begin{bmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}\\{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}\\{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}\end{bmatrix}}}
1043:
3911:
818:
2399:
1768:
2183:
1481:
are classified into
Euclidean, spherical, and hyperbolic cases according to whether an associated sum of unit fractions is equal to one, greater than one, or less than one respectively.
2578:
1868:
2791:
1971:
2908:
2879:
2744:
2715:
166:
2573:
1402:
1510:
4046:
2946:
2426:
4605:
3221:
3192:
3120:
3013:
2985:
2525:
2497:
2120:
602:
242:
1352:
1329:
1306:
861:
3964:
3164:
3090:
2447:
2087:
1601:
1103:
1083:
1063:
991:
971:
951:
881:
838:
753:
733:
709:
689:
669:
645:
622:
574:
554:
214:
186:
258:
465:
398:
331:
3940:
4308:; Jacobs, Victoria R.; Jessup, Naomi A.; Hewitt, Amy; Pynes, D'Anna; Krause, Gladys (April 2020), "Unit fractions as superheroes for instruction",
4350:
Wilson, P. Holt; Edgington, Cynthia P.; Nguyen, Kenny H.; Pescosolido, Ryan C.; Confrey, Jere (November 2011), "Fractions: how to fair share",
4572:
4545:
4417:
4393:
3777:
3589:
3554:
3497:
3373:
1407:
1244:{\displaystyle {\frac {4}{5}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{20}}={\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{10}}.}
886:
1752:{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots =\ln 2.}
996:
4598:
3408:
2796:
762:
2336:
4237:
4142:
3969:
3860:
3809:
3729:
3463:
3071:. This states that, for many observed phenomena involving the selection of items from an ordered sequence, the probability that the
75:. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When an object is divided into equal parts, each part is a unit fraction of the whole.
1285:
A pattern of spherical triangles with reflection symmetry across each triangle edge. Spherical reflection patterns like this with
4812:
2125:
3519:
3057:
1271:
4822:
4106:
4055:
1604:
3531:, Cambridge Monographs on Applied and Computational Mathematics, vol. 18, Cambridge University Press, pp. 65–68,
3247:, on a logarithmic scale. The frequencies of the emission lines are proportional to differences of pairs of unit fractions.
4591:
1504:
3295:
533:
88:
1267:
1490:
1122:
3994:
2852:. However, some pairs of fractions whose difference is a unit fraction are not adjacent in this sense: for instance,
2451:
Fibonacci number. He calls this matrix the
Filbert matrix and it has the same property of having an integer inverse.
141:
4018:
3453:
3131:
1127:
Any positive rational number can be written as the sum of distinct unit fractions, in multiple ways. For example,
119:
2534:
4771:
3061:
115:
3259:, proportional to the differences of two unit fractions. An explanation for this phenomenon is provided by the
3244:
1113:
Several constructions in mathematics involve combining multiple unit fractions together, often by adding them.
123:
1762:
1275:
17:
1607:. Changing every other addition to a subtraction produces the alternating harmonic series, which sums to the
4759:
3282:
756:
712:
4489:
3231:
method does exactly this, and then packs each bin using items of only a single rounded unit fraction size.
3142:
One motivation for this is as a test case for more general bin packing methods. Another involves a form of
4641:
4014:
3272:
2749:
1608:
1474:
520:
As the last of these formulas shows, every fraction can be expressed as a quotient of two unit fractions.
325:
193:
64:
1954:
3609:
3417:
3307:
3033:
1852:{\displaystyle 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}.}
107:
4754:
4212:
3923:
3441:
3228:
2884:
2855:
2720:
2691:
1496:
3666:
3328:
Cavey, Laurie O.; Kinzel, Margaret T. (February 2014), "From whole numbers to invert and multiply",
4817:
4193:
3604:
3481:
3143:
4371:
2959:
at a given fraction and have the squared denominator of the fraction as their diameter. Fractions
1281:
4717:
4702:
4560:
4533:
4512:
4468:
4367:
4333:
4325:
4254:
4202:
4159:
4123:
4072:
3986:
3893:
3877:
3826:
3746:
3707:
3681:
3644:
3618:
3560:
3532:
3406:
Humenberger, Hans (Fall 2014), "Egyptian fractions – representations as sums of unit fractions",
3345:
2681:{\displaystyle \left|{\frac {1}{a}}-{\frac {1}{b}}\right|={\frac {|ad-bc|}{bd}}={\frac {1}{bd}}.}
1357:
529:
111:
4022:
4827:
4732:
4568:
4541:
4436:
4413:
4389:
4388:, Wiley Series in Probability and Statistics, vol. 246, John Wiley and Sons, p. 66,
3800:
3773:
3585:
3550:
3523:
3493:
3459:
3369:
3363:
3349:
1958:
1946:{\displaystyle 1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+\cdots ={\frac {\pi ^{2}}{6}}.}
1580:
1255:
84:
3767:
2913:
4776:
4504:
4452:
4359:
4317:
4288:
4246:
4151:
4115:
4064:
3978:
3932:
3869:
3818:
3738:
3691:
3628:
3542:
3515:
3485:
3437:
3337:
3304:, a number that produces a unit fraction when used as the numerator with a given denominator
3278:
2330:
1965:
648:
4464:
4266:
4171:
3889:
3838:
3787:
3703:
3640:
3365:
The Math We Need to Know and Do in Grades 6 9: Concepts, Skills, Standards, and
Assessments
2404:
110:
as an early step toward the understanding of other fractions. Unit fractions are common in
4766:
4727:
4460:
4262:
4167:
3885:
3834:
3783:
3699:
3636:
3256:
3016:
1259:
135:
80:
3198:
3169:
3097:
2990:
2962:
2502:
2474:
2097:
2045:{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =2.}
579:
219:
83:
can be represented as a sum of distinct unit fractions; these representations are called
4216:
3881:
1334:
1311:
1288:
843:
4412:, Lecture Notes in Economics and Mathematical Systems, vol. 632, Springer-Verlag,
4363:
4232:
4097:
3763:
3724:
3577:
3449:
3264:
3149:
3075:
2432:
2322:
2072:
2061:
1586:
1478:
1088:
1068:
1048:
976:
956:
936:
866:
823:
738:
718:
694:
674:
654:
630:
607:
559:
539:
252:
199:
189:
171:
95:
72:
4329:
4806:
4712:
4337:
4127:
4076:
3268:
3068:
3037:
2065:
1862:
1263:
103:
4516:
4185:
3897:
3711:
3648:
1258:, because the ancient Egyptian civilisations used them as notation for more general
4742:
4737:
4707:
4472:
4305:
4292:
4250:
3936:
3822:
3341:
3271:
are inversely proportional to square unit fractions, and the energy of a photon is
3040:
are a standard classroom example in teaching students to work with unit fractions.
2459:
2091:
1572:{\displaystyle {\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}}
3564:
1507:, the sum of all positive unit fractions. This sum diverges, and its partial sums
3667:"Counting the number of solutions to the Erdős–Straus equation on unit fractions"
3546:
3421:
3804:
3662:
3632:
3301:
3239:
3135:
2956:
2952:
2464:
536:. This conversion can be used to perform modular division: dividing by a number
321:
99:
68:
33:
3852:
532:, any unit fraction can be converted into an equivalent whole number using the
4786:
4781:
4529:
4440:
3695:
3445:
3260:
3873:
4628:
4508:
4456:
328:
two unit fractions produces a result that is generally not a unit fraction:
56:
4321:
3064:, probabilities of this form arise frequently in statistical calculations.
255:
any two unit fractions results in a product that is another unit fraction:
39:
3772:, Pure and Applied Mathematics, vol. 61, Academic Press, p. 65,
2329:
defined a matrix whose elements are unit fractions whose denominators are
2321:
is a
Hilbert matrix. It has the unusual property that all elements in its
4649:
4614:
4080:
3255:
that can be absorbed or emitted by a hydrogen atom are, according to the
317:
94:
In geometry, unit fractions can be used to characterize the curvature of
52:
4207:
3853:"On Riemann's rearrangement theorem for the alternating harmonic series"
3623:
4749:
4722:
4645:
4583:
4258:
4163:
4119:
4068:
3990:
3830:
3750:
3048:
91:. Many infinite sums of unit fractions are meaningful mathematically.
4443:(2007), "Windows scheduling as a restricted version of bin packing",
4019:"A proof that Euler missed ... Apéry's proof of the irrationality of
3252:
4155:
3982:
3742:
1462:{\displaystyle {\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}>1}
309:{\displaystyle {\frac {1}{x}}\times {\frac {1}{y}}={\frac {1}{xy}}.}
4140:
Choi, Man Duen (1983), "Tricks or treats with the
Hilbert matrix",
2575:
which implies that they differ from each other by a unit fraction:
4618:
4565:
Eddington's Search for a
Fundamental Theory: A Key to the Universe
4408:
Saichev, Alexander; Malevergne, Yannick; Sornette, Didier (2009),
3686:
3537:
3238:
3047:
2910:
differ by a unit fraction, but are not adjacent, because for them
2458:
1280:
38:
4279:
Polkinghorne, Ada R. (May 1935), "Young-children and fractions",
3166:
times the length of a message must occupy a fraction of at least
511:{\displaystyle {\frac {1}{x}}\div {\frac {1}{y}}={\frac {y}{x}}.}
2845:{\displaystyle {\tfrac {3}{5}}-{\tfrac {1}{2}}={\tfrac {1}{10}}}
4587:
4490:"SIGACT news online algorithms column 20: The power of harmony"
454:{\displaystyle {\frac {1}{x}}-{\frac {1}{y}}={\frac {y-x}{xy}}}
387:{\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {x+y}{xy}}}
3458:(2nd ed.), MIT Press and McGraw-Hill, pp. 869–872,
3488:(2015), "Section 24.2.2: Modular multiplicative inverses",
3052:
A six-sided die has probability 1/6 of landing on each side
1274:
concern sums of unit fractions, as does the definition of
3607:(2003), "On a coloring conjecture about unit fractions",
3067:
Unequal probabilities related to unit fractions arise in
3060:, all probabilities are equal unit fractions. Due to the
1085:) can instead be performed by multiplying by the integer
196:
of the positive integers. When something is divided into
3310:, one plus a unit fraction, important in musical harmony
2955:. These are a system of circles that are tangent to the
1491:
List of sums of reciprocals § Infinitely many terms
3727:(1948), "On the averages of the divisors of a number",
3298:, a puzzle involving fair division into unit fractions
3094:
item is selected is proportional to the unit fraction
2889:
2860:
2831:
2816:
2801:
2725:
2696:
2197:
1442:
1427:
1412:
1123:
List of sums of reciprocals § Finitely many terms
4025:
3201:
3172:
3152:
3100:
3078:
2993:
2965:
2916:
2887:
2858:
2799:
2752:
2723:
2694:
2581:
2537:
2505:
2477:
2435:
2407:
2339:
2191:
2128:
2100:
2075:
1974:
1871:
1771:
1617:
1589:
1513:
1410:
1360:
1337:
1314:
1291:
1136:
1091:
1071:
1051:
999:
979:
959:
939:
890:
889:
869:
846:
826:
766:
765:
741:
721:
697:
677:
657:
633:
610:
582:
562:
542:
468:
401:
334:
261:
222:
202:
174:
144:
926:{\displaystyle \displaystyle ax\equiv 1{\pmod {y}}.}
4667:
4634:
3584:(3rd ed.), Springer-Verlag, pp. 252–262,
3015:are adjacent if and only if their Ford circles are
1499:have terms that are unit fractions. These include:
576:, can be performed by converting the unit fraction
122:and in analyzing the pattern of frequencies in the
4040:
3452:(2001) , "31.4 Solving modular linear equations",
3215:
3186:
3158:
3114:
3084:
3007:
2979:
2940:
2902:
2873:
2844:
2785:
2738:
2709:
2680:
2567:
2519:
2491:
2441:
2420:
2393:
2313:
2177:
2114:
2081:
2044:
1945:
1851:
1751:
1595:
1571:
1461:
1396:
1346:
1323:
1300:
1243:
1097:
1077:
1057:
1038:{\displaystyle a\equiv {\frac {1}{x}}{\pmod {y}}.}
1037:
985:
965:
945:
925:
875:
855:
832:
812:
747:
727:
703:
683:
663:
639:
616:
596:
568:
548:
510:
453:
386:
308:
236:
208:
180:
160:
813:{\displaystyle \displaystyle ax+by=\gcd(x,y)=1.}
785:
3807:(1971), "Partial sums of the harmonic series",
2394:{\displaystyle C_{i,j}={\frac {1}{F_{i+j-1}}},}
1865:concerns the sum of the square unit fractions:
3674:Journal of the Australian Mathematical Society
3522:(2010), "2.5 Modular division and inversion",
18:Any rational number is a sum of unit fractions
4599:
8:
711:). The extended Euclidean algorithm for the
2178:{\displaystyle B_{i,j}={\frac {1}{i+j-1}}.}
106:, and this familiar application is used in
4606:
4592:
4584:
4483:
4481:
3769:Noneuclidean Tesselations and their Groups
3263:, according to which the energy levels of
2326:
71:of the fraction, which must be a positive
4352:Mathematics Teaching in the Middle School
4206:
4024:
3912:"The discovery of the series formula for
3685:
3622:
3536:
3205:
3200:
3176:
3171:
3151:
3104:
3099:
3077:
2997:
2992:
2969:
2964:
2951:This terminology comes from the study of
2915:
2888:
2886:
2859:
2857:
2830:
2815:
2800:
2798:
2751:
2724:
2722:
2695:
2693:
2660:
2641:
2621:
2618:
2600:
2587:
2580:
2536:
2509:
2504:
2481:
2476:
2434:
2412:
2406:
2368:
2359:
2344:
2338:
2293:
2281:
2269:
2255:
2243:
2231:
2217:
2205:
2192:
2190:
2148:
2133:
2127:
2104:
2099:
2074:
2020:
2007:
1994:
1981:
1973:
1929:
1923:
1904:
1891:
1878:
1870:
1836:
1817:
1804:
1791:
1778:
1770:
1721:
1708:
1695:
1682:
1655:
1639:
1633:
1622:
1616:
1588:
1559:
1540:
1527:
1514:
1512:
1441:
1426:
1411:
1409:
1359:
1336:
1313:
1290:
1228:
1215:
1202:
1189:
1176:
1163:
1150:
1137:
1135:
1090:
1070:
1050:
1016:
1006:
998:
978:
958:
938:
903:
888:
868:
845:
825:
764:
740:
720:
696:
676:
656:
632:
609:
586:
581:
561:
541:
495:
482:
469:
467:
428:
415:
402:
400:
361:
348:
335:
333:
288:
275:
262:
260:
226:
221:
201:
173:
145:
143:
32:For fractions of a measurement unit, see
3476:
3474:
3432:
3430:
3058:uniform distribution on a discrete space
4227:
4225:
3510:
3508:
3320:
3028:Fair division and mathematics education
863:can be eliminated as it is zero modulo
624:, and then multiplying by that number.
604:into an equivalent whole number modulo
4430:
4428:
3386:
3384:
3275:to the difference between two levels.
1961:, the sum of the cubed unit fractions.
102:. Unit fractions are commonly used in
43:Slices of approximately 1/8 of a pizza
973:, the number that when multiplied by
7:
4540:, World Scientific, pp. 81–86,
2786:{\displaystyle 1\cdot 5-2\cdot 3=-1}
3916:by Leibniz, Gregory and Nilakantha"
3580:(2004), "D11. Egyptian Fractions",
1024:
911:
4372:10.5951/mathteacmiddscho.17.4.0230
4364:10.5951/mathteacmiddscho.17.4.0230
3582:Unsolved problems in number theory
3409:Mathematics and Computer Education
1634:
25:
4538:Modern Atomic and Nuclear Physics
4238:The American Mathematical Monthly
4143:The American Mathematical Monthly
3970:The American Mathematical Monthly
3861:The American Mathematical Monthly
3810:The American Mathematical Monthly
3730:The American Mathematical Monthly
3490:Algorithm Design and Applications
118:. They also have applications in
4658:
4627:
4386:Aspects of Statistical Inference
1354:triangles at each vertex (here,
138:that can be written in the form
4410:Theory of Zipf's Law and Beyond
3851:Freniche, Francisco J. (2010),
2903:{\displaystyle {\tfrac {2}{3}}}
2874:{\displaystyle {\tfrac {1}{3}}}
2739:{\displaystyle {\tfrac {3}{5}}}
2710:{\displaystyle {\tfrac {1}{2}}}
1017:
904:
161:{\displaystyle {\frac {1}{n}},}
4567:, Cambridge University Press,
4445:ACM Transactions on Algorithms
4293:10.1080/00094056.1935.10725374
4251:10.1080/00029890.1938.11990863
4184:Richardson, Thomas M. (2001),
4107:The Mathematical Intelligencer
4056:The Mathematical Intelligencer
4035:
4029:
3937:10.1080/0025570X.1990.11977541
3823:10.1080/00029890.1971.11992881
3350:10.5951/teacchilmath.20.6.0374
3342:10.5951/teacchilmath.20.6.0374
2642:
2622:
1652:
1642:
1028:
1018:
915:
905:
800:
788:
1:
3965:"Euler and the zeta function"
3393:Algebra for Today, First Year
3368:, Corwin Press, p. 157,
3330:Teaching Children Mathematics
2527:(in lowest terms) are called
2068:in which the elements on the
715:can be used to find integers
627:In more detail, suppose that
3547:10.1017/cbo9780511921698.001
3362:Solomon, Pearl Gold (2007),
3296:17-animal inheritance puzzle
2568:{\displaystyle ad-bc=\pm 1,}
2094:all equal the unit fraction
993:produces one. Equivalently,
534:extended Euclidean algorithm
216:equal parts, each part is a
89:ancient Egyptian mathematics
4488:van Stee, Rob (June 2012),
3633:10.4007/annals.2003.157.545
3492:, Wiley, pp. 697–698,
2122:. That is, it has elements
1397:{\displaystyle x,y,z=2,3,5}
134:The unit fractions are the
4844:
3525:Modern Computer Arithmetic
3455:Introduction to Algorithms
3132:combinatorial optimization
3126:Combinatorial optimization
3044:Probability and statistics
2455:Adjacency and Ford circles
1488:
1120:
953:is the modular inverse of
120:combinatorial optimization
31:
4698:
4656:
4625:
4041:{\displaystyle \zeta (3)}
3696:10.1017/S1446788712000468
3062:principle of indifference
2467:differ by a unit fraction
2325:are integers. Similarly,
1605:Euler–Mascheroni constant
116:principle of indifference
4561:Kilmister, Clive William
4100:(September 1983), "From
3882:10.4169/000298910x485969
3874:10.4169/000298910X485969
3245:hydrogen spectral series
2185:For example, the matrix
1579:closely approximate the
671:(otherwise, division by
124:hydrogen spectral series
4813:Fractions (mathematics)
4509:10.1145/2261417.2261440
4457:10.1145/1273340.1273344
4384:Welsh, Alan H. (1996),
4310:The Mathematics Teacher
4015:van der Poorten, Alfred
3963:Ayoub, Raymond (1974),
3283:fine-structure constant
2941:{\displaystyle ad-bc=3}
2463:Fractions with tangent
1272:Erdős–Straus conjecture
713:greatest common divisor
244:fraction of the whole.
194:multiplicative inverses
27:One over a whole number
4330:10.5951/mtlt.2018.0024
4322:10.5951/mtlt.2018.0024
4104:", Old Intelligencer,
4042:
3391:Betz, William (1957),
3248:
3217:
3188:
3160:
3116:
3086:
3053:
3009:
2981:
2942:
2904:
2875:
2846:
2787:
2740:
2711:
2682:
2569:
2521:
2493:
2468:
2443:
2422:
2395:
2315:
2179:
2116:
2083:
2046:
1947:
1853:
1753:
1638:
1609:natural logarithm of 2
1597:
1573:
1475:geometric group theory
1470:
1463:
1398:
1348:
1325:
1302:
1276:Ore's harmonic numbers
1254:These sums are called
1245:
1099:
1079:
1059:
1039:
987:
967:
947:
927:
877:
857:
834:
814:
749:
729:
705:
691:is not defined modulo
685:
665:
641:
618:
598:
570:
550:
512:
455:
388:
310:
238:
210:
182:
162:
98:and the tangencies of
87:based on their use in
65:multiplicative inverse
44:
4823:Elementary arithmetic
4235:(1938), "Fractions",
4043:
3661:Elsholtz, Christian;
3610:Annals of Mathematics
3442:Leiserson, Charles E.
3308:Superparticular ratio
3251:The energy levels of
3242:
3218:
3189:
3161:
3117:
3087:
3051:
3034:mathematics education
3010:
2982:
2943:
2905:
2876:
2847:
2788:
2741:
2712:
2683:
2570:
2522:
2494:
2462:
2444:
2423:
2421:{\displaystyle F_{i}}
2396:
2316:
2180:
2117:
2084:
2047:
1948:
1854:
1763:Leibniz formula for π
1754:
1618:
1598:
1574:
1464:
1399:
1349:
1326:
1303:
1284:
1246:
1100:
1080:
1060:
1040:
988:
968:
948:
928:
878:
858:
840:arithmetic, the term
835:
815:
750:
730:
706:
686:
666:
642:
619:
599:
571:
551:
513:
456:
389:
311:
248:Elementary arithmetic
239:
211:
183:
163:
108:mathematics education
42:
4186:"The Filbert matrix"
4023:
3924:Mathematics Magazine
3910:Roy, Ranjan (1990),
3605:Croot, Ernest S. III
3482:Goodrich, Michael T.
3229:harmonic bin packing
3199:
3170:
3150:
3098:
3076:
2991:
2963:
2914:
2885:
2856:
2797:
2750:
2721:
2692:
2579:
2535:
2503:
2475:
2433:
2405:
2337:
2189:
2126:
2098:
2073:
1972:
1869:
1769:
1615:
1587:
1511:
1408:
1358:
1335:
1312:
1289:
1268:Erdős–Graham problem
1266:; for instance, the
1134:
1089:
1069:
1049:
997:
977:
957:
937:
887:
867:
844:
824:
763:
739:
719:
695:
675:
655:
631:
608:
580:
560:
540:
466:
399:
332:
259:
220:
200:
192:. They are thus the
188:can be any positive
172:
142:
67:(reciprocal) of the
4534:Hamilton, Joseph H.
4451:(3): A28:1–A28:22,
4306:Empson, Susan Baker
4281:Childhood Education
4217:1999math......5079R
4194:Fibonacci Quarterly
4102:Elements of Algebra
3395:, Ginn, p. 370
3216:{\displaystyle 1/k}
3187:{\displaystyle 1/k}
3144:pinwheel scheduling
3115:{\displaystyle 1/n}
3008:{\displaystyle c/d}
2980:{\displaystyle a/b}
2520:{\displaystyle c/d}
2492:{\displaystyle a/b}
2115:{\displaystyle 1/i}
597:{\displaystyle 1/x}
237:{\displaystyle 1/n}
4635:Division and ratio
4437:Ladner, Richard E.
4120:10.1007/bf03026580
4069:10.1007/BF03028234
4038:
3249:
3213:
3184:
3156:
3112:
3082:
3054:
3005:
2977:
2938:
2900:
2898:
2871:
2869:
2842:
2840:
2825:
2810:
2783:
2736:
2734:
2707:
2705:
2678:
2565:
2517:
2489:
2469:
2439:
2418:
2391:
2311:
2305:
2175:
2112:
2079:
2042:
1943:
1849:
1749:
1593:
1569:
1471:
1459:
1451:
1436:
1421:
1404:) only exist when
1394:
1347:{\displaystyle 2z}
1344:
1324:{\displaystyle 2y}
1321:
1301:{\displaystyle 2x}
1298:
1256:Egyptian fractions
1241:
1095:
1075:
1055:
1035:
983:
963:
943:
923:
922:
873:
856:{\displaystyle by}
853:
830:
810:
809:
745:
725:
701:
681:
661:
637:
614:
594:
566:
546:
530:modular arithmetic
524:Modular arithmetic
508:
451:
384:
306:
234:
206:
178:
158:
112:probability theory
85:Egyptian fractions
45:
4800:
4799:
4574:978-0-521-37165-0
4547:978-981-283-678-6
4419:978-3-642-02945-5
4395:978-0-471-11591-5
3805:Wrench, J. W. Jr.
3779:978-0-08-087377-0
3591:978-0-387-20860-2
3556:978-1-139-49228-7
3516:Brent, Richard P.
3499:978-1-118-33591-8
3486:Tamassia, Roberto
3446:Rivest, Ronald L.
3438:Cormen, Thomas H.
3375:978-1-4129-1726-1
3265:electron orbitals
3159:{\displaystyle k}
3085:{\displaystyle n}
2897:
2868:
2839:
2824:
2809:
2733:
2704:
2673:
2655:
2608:
2595:
2442:{\displaystyle i}
2386:
2331:Fibonacci numbers
2327:Richardson (2001)
2301:
2289:
2277:
2263:
2251:
2239:
2225:
2213:
2170:
2082:{\displaystyle i}
2028:
2015:
2002:
1989:
1959:irrational number
1938:
1912:
1899:
1886:
1844:
1825:
1812:
1799:
1786:
1729:
1716:
1703:
1690:
1671:
1596:{\displaystyle n}
1581:natural logarithm
1567:
1548:
1535:
1522:
1450:
1435:
1420:
1236:
1223:
1210:
1197:
1184:
1171:
1158:
1145:
1098:{\displaystyle a}
1078:{\displaystyle y}
1058:{\displaystyle x}
1045:Thus division by
1014:
986:{\displaystyle x}
966:{\displaystyle x}
946:{\displaystyle a}
876:{\displaystyle y}
833:{\displaystyle y}
757:Bézout's identity
748:{\displaystyle b}
728:{\displaystyle a}
704:{\displaystyle y}
684:{\displaystyle x}
664:{\displaystyle y}
640:{\displaystyle x}
617:{\displaystyle y}
569:{\displaystyle y}
549:{\displaystyle x}
503:
490:
477:
449:
423:
410:
382:
356:
343:
301:
283:
270:
209:{\displaystyle n}
181:{\displaystyle n}
153:
16:(Redirected from
4835:
4777:Musical interval
4690:
4689:
4687:
4686:
4683:
4680:
4662:
4661:
4631:
4608:
4601:
4594:
4585:
4578:
4577:
4557:
4551:
4550:
4526:
4520:
4519:
4494:
4485:
4476:
4475:
4435:Bar-Noy, Amotz;
4432:
4423:
4422:
4405:
4399:
4398:
4381:
4375:
4374:
4347:
4341:
4340:
4302:
4296:
4295:
4276:
4270:
4269:
4229:
4220:
4219:
4210:
4190:
4181:
4175:
4174:
4137:
4131:
4130:
4094:
4088:
4087:
4085:
4079:, archived from
4052:
4047:
4045:
4044:
4039:
4011:
4005:
4004:
4003:
4002:
3993:, archived from
3960:
3954:
3953:
3952:
3951:
3945:
3939:, archived from
3920:
3915:
3907:
3901:
3900:
3857:
3848:
3842:
3841:
3797:
3791:
3790:
3760:
3754:
3753:
3721:
3715:
3714:
3689:
3671:
3658:
3652:
3651:
3626:
3601:
3595:
3594:
3574:
3568:
3567:
3540:
3530:
3520:Zimmermann, Paul
3512:
3503:
3502:
3478:
3469:
3468:
3434:
3425:
3424:
3403:
3397:
3396:
3388:
3379:
3378:
3359:
3353:
3352:
3325:
3281:argued that the
3279:Arthur Eddington
3222:
3220:
3219:
3214:
3209:
3193:
3191:
3190:
3185:
3180:
3165:
3163:
3162:
3157:
3130:In the study of
3121:
3119:
3118:
3113:
3108:
3093:
3091:
3089:
3088:
3083:
3014:
3012:
3011:
3006:
3001:
2986:
2984:
2983:
2978:
2973:
2947:
2945:
2944:
2939:
2909:
2907:
2906:
2901:
2899:
2890:
2880:
2878:
2877:
2872:
2870:
2861:
2851:
2849:
2848:
2843:
2841:
2832:
2826:
2817:
2811:
2802:
2792:
2790:
2789:
2784:
2745:
2743:
2742:
2737:
2735:
2726:
2716:
2714:
2713:
2708:
2706:
2697:
2687:
2685:
2684:
2679:
2674:
2672:
2661:
2656:
2654:
2646:
2645:
2625:
2619:
2614:
2610:
2609:
2601:
2596:
2588:
2574:
2572:
2571:
2566:
2526:
2524:
2523:
2518:
2513:
2498:
2496:
2495:
2490:
2485:
2450:
2448:
2446:
2445:
2440:
2427:
2425:
2424:
2419:
2417:
2416:
2400:
2398:
2397:
2392:
2387:
2385:
2384:
2360:
2355:
2354:
2320:
2318:
2317:
2312:
2310:
2309:
2302:
2294:
2290:
2282:
2278:
2270:
2264:
2256:
2252:
2244:
2240:
2232:
2226:
2218:
2214:
2206:
2184:
2182:
2181:
2176:
2171:
2169:
2149:
2144:
2143:
2121:
2119:
2118:
2113:
2108:
2090:
2088:
2086:
2085:
2080:
2051:
2049:
2048:
2043:
2029:
2021:
2016:
2008:
2003:
1995:
1990:
1982:
1966:geometric series
1955:Apéry's constant
1952:
1950:
1949:
1944:
1939:
1934:
1933:
1924:
1913:
1905:
1900:
1892:
1887:
1879:
1858:
1856:
1855:
1850:
1845:
1837:
1826:
1818:
1813:
1805:
1800:
1792:
1787:
1779:
1758:
1756:
1755:
1750:
1730:
1722:
1717:
1709:
1704:
1696:
1691:
1683:
1672:
1667:
1666:
1665:
1640:
1637:
1632:
1602:
1600:
1599:
1594:
1578:
1576:
1575:
1570:
1568:
1560:
1549:
1541:
1536:
1528:
1523:
1515:
1495:Many well-known
1468:
1466:
1465:
1460:
1452:
1443:
1437:
1428:
1422:
1413:
1403:
1401:
1400:
1395:
1353:
1351:
1350:
1345:
1330:
1328:
1327:
1322:
1307:
1305:
1304:
1299:
1260:rational numbers
1250:
1248:
1247:
1242:
1237:
1229:
1224:
1216:
1211:
1203:
1198:
1190:
1185:
1177:
1172:
1164:
1159:
1151:
1146:
1138:
1104:
1102:
1101:
1096:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1044:
1042:
1041:
1036:
1031:
1015:
1007:
992:
990:
989:
984:
972:
970:
969:
964:
952:
950:
949:
944:
932:
930:
929:
924:
918:
882:
880:
879:
874:
862:
860:
859:
854:
839:
837:
836:
831:
819:
817:
816:
811:
754:
752:
751:
746:
734:
732:
731:
726:
710:
708:
707:
702:
690:
688:
687:
682:
670:
668:
667:
662:
649:relatively prime
646:
644:
643:
638:
623:
621:
620:
615:
603:
601:
600:
595:
590:
575:
573:
572:
567:
555:
553:
552:
547:
517:
515:
514:
509:
504:
496:
491:
483:
478:
470:
460:
458:
457:
452:
450:
448:
440:
429:
424:
416:
411:
403:
393:
391:
390:
385:
383:
381:
373:
362:
357:
349:
344:
336:
315:
313:
312:
307:
302:
300:
289:
284:
276:
271:
263:
243:
241:
240:
235:
230:
215:
213:
212:
207:
187:
185:
184:
179:
167:
165:
164:
159:
154:
146:
136:rational numbers
62:
55:with one as its
21:
4843:
4842:
4838:
4837:
4836:
4834:
4833:
4832:
4803:
4802:
4801:
4796:
4767:Just intonation
4694:
4684:
4681:
4678:
4677:
4675:
4674:
4663:
4659:
4654:
4632:
4621:
4612:
4582:
4581:
4575:
4559:
4558:
4554:
4548:
4528:
4527:
4523:
4497:ACM SIGACT News
4492:
4487:
4486:
4479:
4434:
4433:
4426:
4420:
4407:
4406:
4402:
4396:
4383:
4382:
4378:
4349:
4348:
4344:
4304:
4303:
4299:
4278:
4277:
4273:
4231:
4230:
4223:
4208:math.RA/9905079
4188:
4183:
4182:
4178:
4156:10.2307/2975779
4139:
4138:
4134:
4098:Euler, Leonhard
4096:
4095:
4091:
4083:
4050:
4021:
4020:
4013:
4012:
4008:
4000:
3998:
3983:10.2307/2319041
3977:(10): 1067–86,
3962:
3961:
3957:
3949:
3947:
3943:
3918:
3913:
3909:
3908:
3904:
3855:
3850:
3849:
3845:
3801:Boas, R. P. Jr.
3799:
3798:
3794:
3780:
3764:Magnus, Wilhelm
3762:
3761:
3757:
3743:10.2307/2305616
3737:(10): 615–619,
3723:
3722:
3718:
3669:
3660:
3659:
3655:
3624:math.NT/0311421
3603:
3602:
3598:
3592:
3578:Guy, Richard K.
3576:
3575:
3571:
3557:
3528:
3514:
3513:
3506:
3500:
3480:
3479:
3472:
3466:
3450:Stein, Clifford
3436:
3435:
3428:
3405:
3404:
3400:
3390:
3389:
3382:
3376:
3361:
3360:
3356:
3327:
3326:
3322:
3317:
3292:
3257:Rydberg formula
3237:
3223:as item sizes.
3197:
3196:
3168:
3167:
3148:
3147:
3128:
3096:
3095:
3074:
3073:
3072:
3046:
3030:
3025:
3017:tangent circles
2989:
2988:
2961:
2960:
2912:
2911:
2883:
2882:
2854:
2853:
2795:
2794:
2748:
2747:
2719:
2718:
2690:
2689:
2665:
2647:
2620:
2586:
2582:
2577:
2576:
2533:
2532:
2501:
2500:
2473:
2472:
2457:
2431:
2430:
2429:
2408:
2403:
2402:
2364:
2340:
2335:
2334:
2304:
2303:
2291:
2279:
2266:
2265:
2253:
2241:
2228:
2227:
2215:
2203:
2193:
2187:
2186:
2153:
2129:
2124:
2123:
2096:
2095:
2071:
2070:
2069:
2058:
1970:
1969:
1925:
1867:
1866:
1767:
1766:
1651:
1641:
1613:
1612:
1585:
1584:
1509:
1508:
1505:harmonic series
1497:infinite series
1493:
1487:
1485:Infinite series
1479:triangle groups
1406:
1405:
1356:
1355:
1333:
1332:
1310:
1309:
1287:
1286:
1132:
1131:
1125:
1119:
1111:
1087:
1086:
1067:
1066:
1047:
1046:
995:
994:
975:
974:
955:
954:
935:
934:
885:
884:
865:
864:
842:
841:
822:
821:
761:
760:
737:
736:
717:
716:
693:
692:
673:
672:
653:
652:
629:
628:
606:
605:
578:
577:
558:
557:
538:
537:
526:
464:
463:
441:
430:
397:
396:
374:
363:
330:
329:
293:
257:
256:
250:
218:
217:
198:
197:
170:
169:
140:
139:
132:
96:triangle groups
81:rational number
60:
37:
28:
23:
22:
15:
12:
11:
5:
4841:
4839:
4831:
4830:
4825:
4820:
4815:
4805:
4804:
4798:
4797:
4795:
4794:
4789:
4784:
4779:
4774:
4769:
4764:
4763:
4762:
4752:
4747:
4746:
4745:
4735:
4730:
4725:
4720:
4715:
4710:
4705:
4699:
4696:
4695:
4693:
4692:
4671:
4669:
4665:
4664:
4657:
4655:
4653:
4652:
4638:
4636:
4633:
4626:
4623:
4622:
4613:
4611:
4610:
4603:
4596:
4588:
4580:
4579:
4573:
4552:
4546:
4521:
4503:(2): 127–136,
4477:
4424:
4418:
4400:
4394:
4376:
4358:(4): 230–236,
4342:
4316:(4): 278–286,
4297:
4287:(8): 354–358,
4271:
4245:(9): 586–601,
4221:
4201:(3): 268–275,
4176:
4150:(5): 301–312,
4132:
4089:
4063:(4): 195–203,
4037:
4034:
4031:
4028:
4006:
3955:
3931:(5): 291–306,
3902:
3868:(5): 442–448,
3843:
3817:(8): 864–870,
3792:
3778:
3755:
3716:
3653:
3617:(2): 545–556,
3596:
3590:
3569:
3555:
3504:
3498:
3470:
3464:
3426:
3416:(3): 268–283,
3398:
3380:
3374:
3354:
3336:(6): 374–383,
3319:
3318:
3316:
3313:
3312:
3311:
3305:
3299:
3291:
3288:
3236:
3233:
3212:
3208:
3204:
3183:
3179:
3175:
3155:
3127:
3124:
3111:
3107:
3103:
3081:
3045:
3042:
3029:
3026:
3024:
3021:
3004:
3000:
2996:
2976:
2972:
2968:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2896:
2893:
2867:
2864:
2838:
2835:
2829:
2823:
2820:
2814:
2808:
2805:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2746:are adjacent:
2732:
2729:
2703:
2700:
2688:For instance,
2677:
2671:
2668:
2664:
2659:
2653:
2650:
2644:
2640:
2637:
2634:
2631:
2628:
2624:
2617:
2613:
2607:
2604:
2599:
2594:
2591:
2585:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2516:
2512:
2508:
2488:
2484:
2480:
2471:Two fractions
2456:
2453:
2438:
2415:
2411:
2390:
2383:
2380:
2377:
2374:
2371:
2367:
2363:
2358:
2353:
2350:
2347:
2343:
2323:inverse matrix
2308:
2300:
2297:
2292:
2288:
2285:
2280:
2276:
2273:
2268:
2267:
2262:
2259:
2254:
2250:
2247:
2242:
2238:
2235:
2230:
2229:
2224:
2221:
2216:
2212:
2209:
2204:
2202:
2199:
2198:
2196:
2174:
2168:
2165:
2162:
2159:
2156:
2152:
2147:
2142:
2139:
2136:
2132:
2111:
2107:
2103:
2078:
2062:Hilbert matrix
2057:
2054:
2053:
2052:
2041:
2038:
2035:
2032:
2027:
2024:
2019:
2014:
2011:
2006:
2001:
1998:
1993:
1988:
1985:
1980:
1977:
1962:
1942:
1937:
1932:
1928:
1922:
1919:
1916:
1911:
1908:
1903:
1898:
1895:
1890:
1885:
1882:
1877:
1874:
1859:
1848:
1843:
1840:
1835:
1832:
1829:
1824:
1821:
1816:
1811:
1808:
1803:
1798:
1795:
1790:
1785:
1782:
1777:
1774:
1759:
1748:
1745:
1742:
1739:
1736:
1733:
1728:
1725:
1720:
1715:
1712:
1707:
1702:
1699:
1694:
1689:
1686:
1681:
1678:
1675:
1670:
1664:
1661:
1658:
1654:
1650:
1647:
1644:
1636:
1631:
1628:
1625:
1621:
1592:
1566:
1563:
1558:
1555:
1552:
1547:
1544:
1539:
1534:
1531:
1526:
1521:
1518:
1486:
1483:
1458:
1455:
1449:
1446:
1440:
1434:
1431:
1425:
1419:
1416:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1343:
1340:
1320:
1317:
1297:
1294:
1252:
1251:
1240:
1235:
1232:
1227:
1222:
1219:
1214:
1209:
1206:
1201:
1196:
1193:
1188:
1183:
1180:
1175:
1170:
1167:
1162:
1157:
1154:
1149:
1144:
1141:
1118:
1115:
1110:
1107:
1094:
1074:
1054:
1034:
1030:
1027:
1023:
1020:
1013:
1010:
1005:
1002:
982:
962:
942:
921:
917:
914:
910:
907:
902:
899:
896:
893:
883:. This leaves
872:
852:
849:
829:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
759:is satisfied:
744:
724:
700:
680:
660:
636:
613:
593:
589:
585:
565:
545:
525:
522:
507:
502:
499:
494:
489:
486:
481:
476:
473:
447:
444:
439:
436:
433:
427:
422:
419:
414:
409:
406:
380:
377:
372:
369:
366:
360:
355:
352:
347:
342:
339:
305:
299:
296:
292:
287:
282:
279:
274:
269:
266:
249:
246:
233:
229:
225:
205:
190:natural number
177:
157:
152:
149:
131:
128:
73:natural number
51:is a positive
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4840:
4829:
4826:
4824:
4821:
4819:
4816:
4814:
4811:
4810:
4808:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4775:
4773:
4770:
4768:
4765:
4761:
4758:
4757:
4756:
4753:
4751:
4748:
4744:
4741:
4740:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4719:
4716:
4714:
4711:
4709:
4706:
4704:
4701:
4700:
4697:
4673:
4672:
4670:
4666:
4651:
4647:
4643:
4640:
4639:
4637:
4630:
4624:
4620:
4616:
4609:
4604:
4602:
4597:
4595:
4590:
4589:
4586:
4576:
4570:
4566:
4562:
4556:
4553:
4549:
4543:
4539:
4535:
4531:
4525:
4522:
4518:
4514:
4510:
4506:
4502:
4498:
4491:
4484:
4482:
4478:
4474:
4470:
4466:
4462:
4458:
4454:
4450:
4446:
4442:
4438:
4431:
4429:
4425:
4421:
4415:
4411:
4404:
4401:
4397:
4391:
4387:
4380:
4377:
4373:
4369:
4365:
4361:
4357:
4353:
4346:
4343:
4339:
4335:
4331:
4327:
4323:
4319:
4315:
4311:
4307:
4301:
4298:
4294:
4290:
4286:
4282:
4275:
4272:
4268:
4264:
4260:
4256:
4252:
4248:
4244:
4240:
4239:
4234:
4228:
4226:
4222:
4218:
4214:
4209:
4204:
4200:
4196:
4195:
4187:
4180:
4177:
4173:
4169:
4165:
4161:
4157:
4153:
4149:
4145:
4144:
4136:
4133:
4129:
4125:
4121:
4117:
4113:
4109:
4108:
4103:
4099:
4093:
4090:
4086:on 2011-07-06
4082:
4078:
4074:
4070:
4066:
4062:
4058:
4057:
4049:
4032:
4026:
4016:
4010:
4007:
3997:on 2019-08-14
3996:
3992:
3988:
3984:
3980:
3976:
3972:
3971:
3966:
3959:
3956:
3946:on 2023-03-14
3942:
3938:
3934:
3930:
3926:
3925:
3917:
3906:
3903:
3899:
3895:
3891:
3887:
3883:
3879:
3875:
3871:
3867:
3863:
3862:
3854:
3847:
3844:
3840:
3836:
3832:
3828:
3824:
3820:
3816:
3812:
3811:
3806:
3802:
3796:
3793:
3789:
3785:
3781:
3775:
3771:
3770:
3765:
3759:
3756:
3752:
3748:
3744:
3740:
3736:
3732:
3731:
3726:
3720:
3717:
3713:
3709:
3705:
3701:
3697:
3693:
3688:
3683:
3680:(1): 50–105,
3679:
3675:
3668:
3664:
3657:
3654:
3650:
3646:
3642:
3638:
3634:
3630:
3625:
3620:
3616:
3612:
3611:
3606:
3600:
3597:
3593:
3587:
3583:
3579:
3573:
3570:
3566:
3562:
3558:
3552:
3548:
3544:
3539:
3534:
3527:
3526:
3521:
3517:
3511:
3509:
3505:
3501:
3495:
3491:
3487:
3483:
3477:
3475:
3471:
3467:
3465:0-262-03293-7
3461:
3457:
3456:
3451:
3447:
3443:
3439:
3433:
3431:
3427:
3423:
3419:
3415:
3411:
3410:
3402:
3399:
3394:
3387:
3385:
3381:
3377:
3371:
3367:
3366:
3358:
3355:
3351:
3347:
3343:
3339:
3335:
3331:
3324:
3321:
3314:
3309:
3306:
3303:
3300:
3297:
3294:
3293:
3289:
3287:
3284:
3280:
3276:
3274:
3270:
3269:hydrogen atom
3266:
3262:
3258:
3254:
3246:
3241:
3234:
3232:
3230:
3224:
3210:
3206:
3202:
3181:
3177:
3173:
3153:
3145:
3140:
3137:
3133:
3125:
3123:
3109:
3105:
3101:
3079:
3070:
3065:
3063:
3059:
3050:
3043:
3041:
3039:
3038:fair division
3035:
3027:
3022:
3020:
3018:
3002:
2998:
2994:
2974:
2970:
2966:
2958:
2954:
2949:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2894:
2891:
2865:
2862:
2836:
2833:
2827:
2821:
2818:
2812:
2806:
2803:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2730:
2727:
2701:
2698:
2675:
2669:
2666:
2662:
2657:
2651:
2648:
2638:
2635:
2632:
2629:
2626:
2615:
2611:
2605:
2602:
2597:
2592:
2589:
2583:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2530:
2514:
2510:
2506:
2486:
2482:
2478:
2466:
2461:
2454:
2452:
2436:
2413:
2409:
2388:
2381:
2378:
2375:
2372:
2369:
2365:
2361:
2356:
2351:
2348:
2345:
2341:
2332:
2328:
2324:
2306:
2298:
2295:
2286:
2283:
2274:
2271:
2260:
2257:
2248:
2245:
2236:
2233:
2222:
2219:
2210:
2207:
2200:
2194:
2172:
2166:
2163:
2160:
2157:
2154:
2150:
2145:
2140:
2137:
2134:
2130:
2109:
2105:
2101:
2093:
2076:
2067:
2066:square matrix
2063:
2055:
2039:
2036:
2033:
2030:
2025:
2022:
2017:
2012:
2009:
2004:
1999:
1996:
1991:
1986:
1983:
1978:
1975:
1967:
1963:
1960:
1956:
1940:
1935:
1930:
1926:
1920:
1917:
1914:
1909:
1906:
1901:
1896:
1893:
1888:
1883:
1880:
1875:
1872:
1864:
1863:Basel problem
1860:
1846:
1841:
1838:
1833:
1830:
1827:
1822:
1819:
1814:
1809:
1806:
1801:
1796:
1793:
1788:
1783:
1780:
1775:
1772:
1764:
1760:
1746:
1743:
1740:
1737:
1734:
1731:
1726:
1723:
1718:
1713:
1710:
1705:
1700:
1697:
1692:
1687:
1684:
1679:
1676:
1673:
1668:
1662:
1659:
1656:
1648:
1645:
1629:
1626:
1623:
1619:
1610:
1606:
1590:
1582:
1564:
1561:
1556:
1553:
1550:
1545:
1542:
1537:
1532:
1529:
1524:
1519:
1516:
1506:
1502:
1501:
1500:
1498:
1492:
1484:
1482:
1480:
1476:
1456:
1453:
1447:
1444:
1438:
1432:
1429:
1423:
1417:
1414:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1341:
1338:
1318:
1315:
1295:
1292:
1283:
1279:
1277:
1273:
1269:
1265:
1264:number theory
1261:
1257:
1238:
1233:
1230:
1225:
1220:
1217:
1212:
1207:
1204:
1199:
1194:
1191:
1186:
1181:
1178:
1173:
1168:
1165:
1160:
1155:
1152:
1147:
1142:
1139:
1130:
1129:
1128:
1124:
1116:
1114:
1108:
1106:
1092:
1072:
1052:
1032:
1025:
1021:
1011:
1008:
1003:
1000:
980:
960:
940:
919:
912:
908:
900:
897:
894:
891:
870:
850:
847:
827:
806:
803:
797:
794:
791:
782:
779:
776:
773:
770:
767:
758:
742:
722:
714:
698:
678:
658:
650:
634:
625:
611:
591:
587:
583:
563:
543:
535:
531:
523:
521:
518:
505:
500:
497:
492:
487:
484:
479:
474:
471:
461:
445:
442:
437:
434:
431:
425:
420:
417:
412:
407:
404:
394:
378:
375:
370:
367:
364:
358:
353:
350:
345:
340:
337:
327:
323:
319:
303:
297:
294:
290:
285:
280:
277:
272:
267:
264:
254:
247:
245:
231:
227:
223:
203:
195:
191:
175:
155:
150:
147:
137:
129:
127:
125:
121:
117:
113:
109:
105:
104:fair division
101:
97:
92:
90:
86:
82:
76:
74:
70:
66:
58:
54:
50:
49:unit fraction
41:
35:
30:
19:
4791:
4564:
4555:
4537:
4524:
4500:
4496:
4448:
4444:
4409:
4403:
4385:
4379:
4355:
4351:
4345:
4313:
4309:
4300:
4284:
4280:
4274:
4242:
4236:
4198:
4192:
4179:
4147:
4141:
4135:
4114:(3): 75–76,
4111:
4105:
4101:
4092:
4081:the original
4060:
4054:
4009:
3999:, retrieved
3995:the original
3974:
3968:
3958:
3948:, retrieved
3941:the original
3928:
3922:
3905:
3865:
3859:
3846:
3814:
3808:
3795:
3768:
3758:
3734:
3728:
3725:Ore, Øystein
3719:
3677:
3673:
3663:Tao, Terence
3656:
3614:
3608:
3599:
3581:
3572:
3524:
3489:
3454:
3413:
3407:
3401:
3392:
3364:
3357:
3333:
3329:
3323:
3277:
3250:
3225:
3141:
3129:
3066:
3055:
3031:
3023:Applications
2953:Ford circles
2950:
2528:
2470:
2465:Ford circles
2428:denotes the
2092:antidiagonal
2059:
1494:
1472:
1253:
1126:
1112:
1109:Combinations
626:
527:
519:
462:
395:
251:
133:
100:Ford circles
93:
77:
63:. It is the
48:
46:
29:
4755:Irreducible
4685:Denominator
4530:Yang, Fujia
4441:Tamir, Tami
4233:Ford, L. R.
3302:Submultiple
3136:bin packing
2957:number line
1964:The binary
1953:Similarly,
1117:Finite sums
322:subtracting
253:Multiplying
114:due to the
69:denominator
34:Unit prefix
4818:1 (number)
4807:Categories
4787:Percentage
4782:Paper size
4691:= Quotient
4001:2023-03-22
3950:2023-03-22
3422:1622317875
3315:References
3261:Bohr model
3134:problems,
3069:Zipf's law
1489:See also:
1121:See also:
820:In modulo-
755:such that
130:Arithmetic
4760:Reduction
4718:Continued
4703:Algebraic
4679:Numerator
4615:Fractions
4338:216283105
4128:122191726
4077:121589323
4027:ζ
3687:1107.1010
3538:1004.4710
3273:quantized
2924:−
2813:−
2778:−
2769:⋅
2763:−
2757:⋅
2633:−
2598:−
2557:±
2545:−
2379:−
2164:−
2034:⋯
1927:π
1918:⋯
1839:π
1831:⋯
1828:−
1802:−
1776:−
1744:
1735:⋯
1732:−
1706:−
1680:−
1646:−
1635:∞
1620:∑
1603:plus the
1554:⋯
1004:≡
933:That is,
898:≡
556:, modulo
480:÷
435:−
413:−
316:However,
273:×
57:numerator
4828:Integers
4733:Egyptian
4668:Fraction
4650:Quotient
4642:Dividend
4563:(1994),
4536:(2009),
4517:14805804
4017:(1979),
3898:20575373
3766:(1974),
3712:17233943
3665:(2013),
3649:13514070
3418:ProQuest
3290:See also
2529:adjacent
2056:Matrices
1270:and the
1065:(modulo
326:dividing
53:fraction
4750:Integer
4723:Decimal
4688:
4676:
4646:Divisor
4473:2461059
4465:2344019
4267:1524411
4259:2302799
4213:Bibcode
4172:0701570
4164:2975779
3991:2319041
3890:2663251
3839:0289994
3831:2316476
3788:0352287
3751:2305616
3704:3101397
3641:1973054
3253:photons
3235:Physics
4743:Silver
4738:Golden
4728:Dyadic
4713:Binary
4708:Aspect
4619:ratios
4571:
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2401:where
1957:is an
1331:, and
318:adding
168:where
4513:S2CID
4493:(PDF)
4469:S2CID
4368:JSTOR
4334:S2CID
4326:JSTOR
4255:JSTOR
4203:arXiv
4189:(PDF)
4160:JSTOR
4124:S2CID
4084:(PDF)
4073:S2CID
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3987:JSTOR
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3894:S2CID
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3856:(PDF)
3827:JSTOR
3747:JSTOR
3708:S2CID
3682:arXiv
3670:(PDF)
3645:S2CID
3619:arXiv
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3533:arXiv
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3346:JSTOR
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3056:In a
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4792:Unit
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2037:=
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2023:1
2018:+
2013:8
2010:1
2005:+
2000:4
1997:1
1992:+
1987:2
1984:1
1979:+
1976:1
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