1948:
2558:
1568:
3520:
1574:
2072:
1227:
3772:
3220:
1943:{\displaystyle {\begin{aligned}\ln 2&={\frac {1}{2}}+{\frac {1}{2\cdot 2^{2}}}+{\frac {1}{3\cdot 2^{3}}}+{\frac {1}{4\cdot 2^{4}}}+{\frac {1}{5\cdot 2^{5}}}+\cdots \\&=\sum _{k=1}^{\infty }{\frac {1}{2^{k}\cdot k}}={\frac {1}{2}}\sum _{k=0}^{\infty }\left\\&={\frac {1}{2}}P{\bigl (}1,2,1,(1){\bigr )}.\end{aligned}}}
2553:{\displaystyle {\begin{aligned}\arctan {\frac {1}{b}}&={\frac {1}{b}}-{\frac {1}{b^{3}3}}+{\frac {1}{b^{5}5}}-{\frac {1}{b^{7}7}}+{\frac {1}{b^{9}9}}+\cdots \\&=\sum _{k=1}^{\infty }\left={\frac {1}{b}}\sum _{k=0}^{\infty }\left\\&={\frac {1}{b}}P\left(1,b^{4},4,\left(1,0,-b^{-2},0\right)\right).\end{aligned}}}
1563:{\displaystyle {\begin{aligned}\ln {\frac {10}{9}}&={\frac {1}{10}}+{\frac {1}{200}}+{\frac {1}{3\ 000}}+{\frac {1}{40\,000}}+{\frac {1}{500\,000}}+\cdots \\&=\sum _{k=1}^{\infty }{\frac {1}{10^{k}\cdot k}}={\frac {1}{10}}\sum _{k=0}^{\infty }\left\\&={\frac {1}{10}}P{\bigl (}1,10,1,(1){\bigr )},\end{aligned}}}
4293:
that are not counted, computers usually perform arithmetic for many bits (32 or 64) and round, and we are only interested in the most significant digit(s). There is a possibility that a particular computation will be akin to failing to add a small number (e.g. 1) to the number 999999999999999, and
4281:
Since only the fractional part is accurate, extracting the wanted digit requires that one removes the integer part of the final sum, multiplies it by 16 and keeps the integer part to "skim off" the hexadecimal digit at the desired position (in theory, the next few digits up to the accuracy of the
3983:
3988:
Since we only care about the fractional part of the sum, we look at our two terms and realise that only the first sum contains terms with an integer part; conversely, the second sum doesn't contain terms with an integer part, since the numerator can never be larger than the denominator for
2943:
4168:
3515:{\displaystyle \pi =4\sum _{k=0}^{\infty }{\frac {1}{\left(16^{k}\right)(8k+1)}}-2\sum _{k=0}^{\infty }{\frac {1}{\left(16^{k}\right)(8k+4)}}-\sum _{k=0}^{\infty }{\frac {1}{\left(16^{k}\right)(8k+5)}}-\sum _{k=0}^{\infty }{\frac {1}{\left(16^{k}\right)(8k+6)}}.}
3547:
2048:
3791:
3997:. Therefore, we need a trick to remove the integer parts, that we don't need, from the terms of the first sum, in order to speed up and increase the precision of the calculations. That trick is to reduce modulo 8
1207:
4276:
2596:
are zero. The discovery of these formulae involves a computer search for such linear combinations after computing the individual sums. The search procedure consists of choosing a range of parameter values for
2654:
4394:
D. J. Broadhurst provides a generalization of the BBP algorithm that may be used to compute a number of other constants in nearly linear time and logarithmic space. Explicit results are given for
809:
2659:
2077:
1579:
1232:
411:
3767:{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{\left(16^{k}\right)(8k+1)}}=\sum _{k=0}^{n}{\frac {1}{\left(16^{k}\right)(8k+1)}}+\sum _{k=n+1}^{\infty }{\frac {1}{\left(16^{k}\right)(8k+1)}}.}
3127:
4007:
1080:
231:
895:
4643:
4610:
4577:
4376:
4540:
4511:
4482:
4450:
4423:
3197:
553:
525:
437:
4689:
4663:
611:
582:
495:
466:
3171:
631:
4196:
Now to complete the calculation, this must be applied to each of the four sums in turn. Once this is done, the four summations are put back into the sum to
1963:
3978:{\displaystyle \sum _{k=0}^{\infty }{\frac {16^{n-k}}{8k+1}}=\sum _{k=0}^{n}{\frac {16^{n-k}}{8k+1}}+\sum _{k=n+1}^{\infty }{\frac {16^{n-k}}{8k+1}}.}
4382:
require increasingly more time to calculate; that is, the "further out" a digit is, the longer it takes BBP to calculate it, just like the standard
3777:
We now multiply by 16, so that the hexadecimal point (the divide between fractional and integer parts of the number) shifts (or remains, if
1091:
2938:{\displaystyle {\begin{aligned}\pi &=\sum _{k=0}^{\infty }\left\\&=P{\bigl (}1,16,8,(4,0,0,-2,-1,-1,0,0){\bigr )},\end{aligned}}}
4206:
4177:
operator always guarantees that only the fractional parts of the terms of the first sum will be kept. To calculate 16 mod (8
5090:
4863:
5065:
49:
292:
using distributed computing. The existence of this formula came as a surprise. It had been widely believed that computing the
651:
4163:{\displaystyle \sum _{k=0}^{n}{\frac {16^{n-k}{\bmod {(}}8k+1)}{8k+1}}+\sum _{k=n+1}^{\infty }{\frac {16^{n-k}}{8k+1}}.}
313:
2954:
911:
62:
4912:
4746:
2641:
2610:
829:
5031:
4712:
634:
4309:
without requiring custom data types having thousands or even millions of digits. The method calculates the
5085:
4704:
4182:
4453:
4543:
4395:
4910:
Bailey, David H.; Borwein, Jonathan M.; Borwein, Peter B.; Plouffe, Simon (1997). "The quest for pi".
4717:
4692:
4336:
with less computational effort than formulas that must calculate all intervening digits, BBP remains
4326:
3530:
5041:
4286:
4193:
product becomes greater than one, the modulus is taken, just as for the running total in each sum.
2592:
as an exponent of 2 or it some other factor-rich value, but where several of the terms of sequence
4937:
4290:
4174:
905:
function leads to a compact notation for some solutions. For example, the original BBP formula:
4615:
4582:
4549:
4343:
4834:
4516:
4487:
4458:
5037:
5027:
5009:"Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)"
4988:"Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)"
4886:
4831:
4428:
4401:
417:
56:, and Plouffe. Before that, it had been published by Plouffe on his own site. The formula is:
17:
3176:
1217:
Some of the simplest formulae of this type that were well known before BBP and for which the
538:
504:
422:
4921:
4765:
4755:
4668:
3208:
2614:
276:(i.e., in base 10). But another formula discovered by Plouffe in 2022 allows extracting the
237:
4933:
4779:
5072:", web page with links to Bailey's code implementing the BBP algorithm, September 8, 2006.
5049:
4929:
4775:
4648:
4337:
4322:
2055:
587:
558:
471:
442:
4956:
4855:
4001: + 1. Our first sum (out of four) to compute the fractional part then becomes:
3150:
4289:, but only having to perform the summation of some middle columns. While there are some
2618:
616:
5079:
53:
45:
4941:
4806:
5050:"A compendium of BBP-type formulas for mathematical constants, updated 15 Aug 2017"
2043:{\displaystyle \ln {\frac {a}{a-1}}=\sum _{k=1}^{\infty }{\frac {1}{a^{k}\cdot k}}}
257:
4760:
4741:
2625:
that adds up those intermediate sums to a well-known constant or perhaps to zero.
4971:
Run times for the BBP algorithm ... increase roughly linearly with the position
4889:
4186:
3200:
245:
284:
in decimal. BBP and BBP-inspired algorithms have been used in projects such as
5069:
4332:
Though the BBP formula can directly calculate the value of any given digit of
498:
4894:
4839:
4770:
645:
A specialization of the general formula that has produced many results is:
3534:
898:
48:
and is named after the authors of the article in which it was published,
5008:
4987:
4925:
1202:{\displaystyle \pi =P{\bigl (}1,16,8,(4,0,0,-2,-1,-1,0,0){\bigr )}.}
3132:
This formula has been shown through a fairly simple proof to equal
5013:
4271:{\displaystyle 4\Sigma _{1}-2\Sigma _{2}-\Sigma _{3}-\Sigma _{4}.}
528:
285:
555:, there is no known systematic algorithm for finding appropriate
2948:
which also reduces to this equivalent ratio of two polynomials:
4742:"On the Rapid Computation of Various Polylogarithmic Constants"
4321: − 1 digits and can use small, efficient data types.
2584: > 1. Many now-discovered formulae are known for
4294:
that the error will propagate to the most significant digit.
4052:
4740:
Bailey, David H.; Borwein, Peter B.; Plouffe, Simon (1997).
2609:, evaluating the sums out to many digits, and then using an
39:
2572:
function mentioned above, the simplest known formula for
804:{\displaystyle P(s,b,m,A)=\sum _{k=0}^{\infty }\left,}
4691:. These results are obtained primarily by the use of
4671:
4651:
4618:
4585:
4552:
4519:
4490:
4461:
4431:
4404:
4346:
4209:
4010:
3794:
3550:
3223:
3179:
3153:
2957:
2657:
2640:
summation formula was found in 1995 by
Plouffe using
2075:
1966:
1577:
1230:
1094:
914:
832:
654:
619:
590:
561:
541:
507:
474:
445:
425:
316:
65:
4860:
4792:
3147:
We would like to define a formula that returns the (
264:) without computing the preceding digits. This does
3207:. A few manipulations are required to implement a
2062:notation can be also generalized to the case where
307:Since its discovery, formulas of the general form:
4683:
4657:
4637:
4604:
4571:
4534:
4505:
4476:
4444:
4417:
4370:
4270:
4162:
3977:
3766:
3514:
3191:
3165:
3121:
2937:
2552:
2042:
1942:
1562:
1201:
1074:
889:
803:
625:
605:
576:
547:
519:
489:
460:
431:
406:{\displaystyle \alpha =\sum _{k=0}^{\infty }\left}
405:
225:
3122:{\displaystyle \pi =\sum _{k=0}^{\infty }\left.}
27:Formula for computing the nth base-16 digit of π
1075:{\displaystyle \pi =\sum _{k=0}^{\infty }\left}
226:{\displaystyle \pi =\sum _{k=0}^{\infty }\left}
4185:algorithm is done at the same loop level, not
4181: + 1) quickly and efficiently, the
2923:
2838:
1928:
1894:
1548:
1514:
1191:
1106:
8:
890:{\displaystyle A=(a_{1},a_{2},\dots ,a_{m})}
4862:. Simon Fraser University. March 21, 1999.
4298:BBP compared to other methods of computing
4282:calculations used would also be accurate).
1221:function leads to a compact notation, are:
4769:
4759:
4670:
4650:
4623:
4617:
4590:
4584:
4557:
4551:
4518:
4489:
4460:
4436:
4430:
4409:
4403:
4378:), whereby successively larger values of
4345:
4259:
4246:
4233:
4217:
4208:
4129:
4123:
4117:
4100:
4055:
4051:
4039:
4032:
4026:
4015:
4009:
3944:
3938:
3932:
3915:
3880:
3874:
3868:
3857:
3822:
3816:
3810:
3799:
3793:
3730:
3716:
3710:
3693:
3655:
3641:
3635:
3624:
3586:
3572:
3566:
3555:
3549:
3478:
3464:
3458:
3447:
3409:
3395:
3389:
3378:
3340:
3326:
3320:
3309:
3268:
3254:
3248:
3237:
3222:
3178:
3152:
3083:
3067:
3051:
3021:
3011:
2999:
2990:
2979:
2968:
2956:
2922:
2921:
2837:
2836:
2792:
2768:
2744:
2720:
2707:
2698:
2687:
2676:
2658:
2656:
2518:
2479:
2451:
2405:
2395:
2371:
2355:
2346:
2335:
2324:
2310:
2281:
2272:
2264:
2255:
2244:
2233:
2201:
2191:
2176:
2166:
2151:
2141:
2126:
2116:
2103:
2086:
2076:
2074:
2025:
2015:
2009:
1998:
1973:
1965:
1927:
1926:
1893:
1892:
1879:
1842:
1830:
1821:
1810:
1799:
1785:
1767:
1757:
1751:
1740:
1711:
1695:
1683:
1667:
1655:
1639:
1627:
1611:
1598:
1578:
1576:
1547:
1546:
1513:
1512:
1499:
1462:
1450:
1441:
1430:
1419:
1405:
1387:
1377:
1371:
1360:
1333:
1324:
1314:
1305:
1284:
1271:
1258:
1241:
1231:
1229:
1190:
1189:
1105:
1104:
1093:
1041:
1017:
993:
969:
956:
947:
936:
925:
913:
878:
859:
846:
831:
784:
758:
752:
746:
735:
723:
714:
703:
692:
653:
618:
589:
560:
540:
506:
473:
444:
424:
366:
358:
349:
338:
327:
315:
192:
168:
144:
120:
107:
98:
87:
76:
64:
4729:
3529:and taking the first sum, we split the
1953:(In fact, this identity holds true for
300:is just as hard as computing the first
5032:Making An Algorithm An Algorithm — BBP
4285:This process is similar to performing
3214:We must first rewrite the formula as:
4955:Bailey, David H. (8 September 2006).
4735:
4733:
2644:. It is also representable using the
531:. Formulas of this form are known as
7:
4805:Gourdon, Xavier (12 February 2003).
4645:, and various products of powers of
416:have been discovered for many other
3140:BBP digit-extraction algorithm for
4256:
4243:
4230:
4214:
4118:
3933:
3811:
3711:
3567:
3459:
3390:
3321:
3249:
2980:
2688:
2611:integer relation-finding algorithm
2336:
2245:
2010:
1811:
1752:
1431:
1372:
1213:Previously known BBP-type formulae
937:
704:
339:
88:
25:
2054:Plouffe was also inspired by the
236:The BBP formula gives rise to a
5044:", weblog post, March 15, 2009.
4866:from the original on 2017-06-10
3525:Now, for a particular value of
633:; such formulas are discovered
288:for calculating many digits of
44:. It was discovered in 1995 by
5034:", weblog post, July 14, 2010.
4529:
4523:
4500:
4494:
4471:
4465:
4365:
4350:
4073:
4056:
3755:
3740:
3680:
3665:
3611:
3596:
3503:
3488:
3434:
3419:
3365:
3350:
3293:
3278:
2918:
2861:
1923:
1917:
1543:
1537:
1186:
1129:
884:
839:
781:
765:
682:
658:
600:
594:
571:
565:
501:with integer coefficients and
484:
478:
455:
449:
392:
386:
378:
372:
32:Bailey–Borwein–Plouffe formula
18:Bailey-Borwein-Plouffe formula
1:
4761:10.1090/S0025-5718-97-00856-9
2588:as an exponent of 2 or 3 and
2564:The search for new equalities
4835:"Digit-Extraction Algorithm"
5107:
4957:"The BBP Algorithm for Pi"
4913:Mathematical Intelligencer
4747:Mathematics of Computation
4638:{\displaystyle \log ^{5}2}
4605:{\displaystyle \log ^{4}2}
4572:{\displaystyle \log ^{3}2}
4371:{\displaystyle O(n\log n)}
4535:{\displaystyle \zeta (x)}
4506:{\displaystyle \zeta (5)}
4477:{\displaystyle \zeta (3)}
5091:Experimental mathematics
5042:Cook’s Class Contains Pi
4807:"N-th Digit Computation"
4713:Experimental mathematics
4445:{\displaystyle \pi ^{4}}
4418:{\displaystyle \pi ^{3}}
4325:found a variant of BBP,
4305:This algorithm computes
252:(and therefore also the
4386:-computing algorithms.
3192:{\displaystyle n\geq 0}
548:{\displaystyle \alpha }
520:{\displaystyle b\geq 2}
432:{\displaystyle \alpha }
248:(hexadecimal) digit of
4685:
4684:{\displaystyle \log 2}
4659:
4639:
4606:
4573:
4536:
4507:
4478:
4446:
4419:
4372:
4317:calculating the first
4272:
4183:modular exponentiation
4164:
4122:
4031:
3979:
3937:
3873:
3815:
3785:-th fractional digit:
3768:
3715:
3640:
3571:
3516:
3463:
3394:
3325:
3253:
3193:
3167:
3123:
2984:
2939:
2692:
2554:
2340:
2249:
2044:
2014:
1944:
1815:
1756:
1564:
1435:
1376:
1203:
1076:
941:
891:
805:
751:
708:
627:
607:
578:
549:
521:
491:
462:
433:
407:
343:
227:
92:
4693:polylogarithm ladders
4686:
4660:
4640:
4607:
4574:
4544:Riemann zeta function
4537:
4508:
4479:
4447:
4420:
4373:
4273:
4189:. When its running 16
4165:
4096:
4011:
3980:
3911:
3853:
3795:
3781:) to the left of the
3769:
3689:
3620:
3551:
3517:
3443:
3374:
3305:
3233:
3194:
3168:
3124:
2964:
2940:
2672:
2621:) to find a sequence
2555:
2320:
2229:
2045:
1994:
1945:
1795:
1736:
1565:
1415:
1356:
1204:
1077:
921:
892:
806:
731:
688:
628:
608:
579:
550:
522:
492:
463:
434:
408:
323:
228:
72:
4669:
4658:{\displaystyle \pi }
4649:
4616:
4583:
4550:
4517:
4488:
4459:
4429:
4402:
4344:
4207:
4008:
3792:
3548:
3221:
3211:using this formula.
3177:
3151:
2955:
2655:
2629:The BBP formula for
2580: = 1, but
2073:
2066:is not an integer):
1964:
1575:
1228:
1092:
912:
830:
652:
617:
606:{\displaystyle q(k)}
588:
577:{\displaystyle p(k)}
559:
539:
505:
490:{\displaystyle q(k)}
472:
461:{\displaystyle p(k)}
443:
423:
314:
272:th decimal digit of
63:
4329:, which is faster.
4287:long multiplication
3166:{\displaystyle n+1}
2056:arctan power series
1085:can be written as:
38:) is a formula for
5070:BBP Code Directory
5007:D. J. Broadhurst,
4986:D. J. Broadhurst,
4926:10.1007/BF03024340
4887:Weisstein, Eric W.
4832:Weisstein, Eric W.
4705:Approximations of
4681:
4655:
4635:
4602:
4569:
4532:
4503:
4474:
4442:
4415:
4396:Catalan's constant
4368:
4268:
4160:
3975:
3764:
3512:
3189:
3163:
3119:
2935:
2933:
2550:
2548:
2040:
1940:
1938:
1560:
1558:
1199:
1072:
887:
826:are integers, and
801:
623:
603:
574:
545:
517:
487:
458:
429:
418:irrational numbers
403:
240:for computing the
223:
5048:Bailey, David H.
5038:Richard J. Lipton
5028:Richard J. Lipton
4793:Plouffe's website
4718:Bellard's formula
4327:Bellard's formula
4155:
4091:
3970:
3906:
3848:
3759:
3684:
3615:
3507:
3438:
3369:
3297:
3105:
3005:
2811:
2787:
2763:
2739:
2713:
2636:The original BBP
2459:
2429:
2390:
2364:
2318:
2300:
2294:
2270:
2211:
2186:
2161:
2136:
2111:
2094:
2058:of the form (the
2038:
1989:
1887:
1858:
1836:
1793:
1780:
1718:
1690:
1662:
1634:
1606:
1507:
1478:
1456:
1413:
1400:
1338:
1319:
1300:
1295:
1279:
1266:
1249:
1060:
1036:
1012:
988:
962:
901:of integers. The
791:
729:
626:{\displaystyle b}
535:. Given a number
533:BBP-type formulas
396:
364:
211:
187:
163:
139:
113:
16:(Redirected from
5098:
5062:
5060:
5059:
5054:
4995:
4984:
4978:
4977:
4968:
4966:
4961:
4952:
4946:
4945:
4907:
4901:
4900:
4899:
4882:
4876:
4875:
4873:
4871:
4852:
4846:
4845:
4844:
4827:
4821:
4820:
4818:
4816:
4811:
4802:
4796:
4790:
4784:
4783:
4773:
4771:2060/19970009337
4763:
4754:(218): 903–913.
4737:
4708:
4690:
4688:
4687:
4682:
4664:
4662:
4661:
4656:
4644:
4642:
4641:
4636:
4628:
4627:
4611:
4609:
4608:
4603:
4595:
4594:
4578:
4576:
4575:
4570:
4562:
4561:
4541:
4539:
4538:
4533:
4512:
4510:
4509:
4504:
4483:
4481:
4480:
4475:
4454:Apéry's constant
4451:
4449:
4448:
4443:
4441:
4440:
4424:
4422:
4421:
4416:
4414:
4413:
4385:
4377:
4375:
4374:
4369:
4335:
4308:
4301:
4277:
4275:
4274:
4269:
4264:
4263:
4251:
4250:
4238:
4237:
4222:
4221:
4199:
4169:
4167:
4166:
4161:
4156:
4154:
4140:
4139:
4124:
4121:
4116:
4092:
4090:
4076:
4060:
4059:
4050:
4049:
4033:
4030:
4025:
3993: >
3984:
3982:
3981:
3976:
3971:
3969:
3955:
3954:
3939:
3936:
3931:
3907:
3905:
3891:
3890:
3875:
3872:
3867:
3849:
3847:
3833:
3832:
3817:
3814:
3809:
3773:
3771:
3770:
3765:
3760:
3758:
3739:
3735:
3734:
3717:
3714:
3709:
3685:
3683:
3664:
3660:
3659:
3642:
3639:
3634:
3616:
3614:
3595:
3591:
3590:
3573:
3570:
3565:
3521:
3519:
3518:
3513:
3508:
3506:
3487:
3483:
3482:
3465:
3462:
3457:
3439:
3437:
3418:
3414:
3413:
3396:
3393:
3388:
3370:
3368:
3349:
3345:
3344:
3327:
3324:
3319:
3298:
3296:
3277:
3273:
3272:
3255:
3252:
3247:
3209:spigot algorithm
3206:
3198:
3196:
3195:
3190:
3172:
3170:
3169:
3164:
3143:
3135:
3128:
3126:
3125:
3120:
3115:
3111:
3110:
3106:
3104:
3088:
3087:
3072:
3071:
3056:
3055:
3042:
3026:
3025:
3012:
3006:
3004:
3003:
2991:
2983:
2978:
2944:
2942:
2941:
2936:
2934:
2927:
2926:
2842:
2841:
2826:
2822:
2818:
2817:
2813:
2812:
2810:
2793:
2788:
2786:
2769:
2764:
2762:
2745:
2740:
2738:
2721:
2714:
2712:
2711:
2699:
2691:
2686:
2639:
2632:
2615:Helaman Ferguson
2575:
2559:
2557:
2556:
2551:
2549:
2542:
2538:
2537:
2533:
2526:
2525:
2484:
2483:
2460:
2452:
2444:
2440:
2436:
2435:
2431:
2430:
2428:
2414:
2413:
2412:
2396:
2391:
2389:
2372:
2365:
2363:
2362:
2347:
2339:
2334:
2319:
2311:
2306:
2302:
2301:
2296:
2295:
2290:
2282:
2273:
2271:
2269:
2268:
2256:
2248:
2243:
2222:
2212:
2210:
2206:
2205:
2192:
2187:
2185:
2181:
2180:
2167:
2162:
2160:
2156:
2155:
2142:
2137:
2135:
2131:
2130:
2117:
2112:
2104:
2095:
2087:
2049:
2047:
2046:
2041:
2039:
2037:
2030:
2029:
2016:
2013:
2008:
1990:
1988:
1974:
1949:
1947:
1946:
1941:
1939:
1932:
1931:
1898:
1897:
1888:
1880:
1872:
1868:
1864:
1863:
1859:
1857:
1843:
1837:
1835:
1834:
1822:
1814:
1809:
1794:
1786:
1781:
1779:
1772:
1771:
1758:
1755:
1750:
1729:
1719:
1717:
1716:
1715:
1696:
1691:
1689:
1688:
1687:
1668:
1663:
1661:
1660:
1659:
1640:
1635:
1633:
1632:
1631:
1612:
1607:
1599:
1569:
1567:
1566:
1561:
1559:
1552:
1551:
1518:
1517:
1508:
1500:
1492:
1488:
1484:
1483:
1479:
1477:
1463:
1457:
1455:
1454:
1442:
1434:
1429:
1414:
1406:
1401:
1399:
1392:
1391:
1378:
1375:
1370:
1349:
1339:
1337:
1325:
1320:
1318:
1306:
1301:
1299:
1293:
1285:
1280:
1272:
1267:
1259:
1250:
1242:
1208:
1206:
1205:
1200:
1195:
1194:
1110:
1109:
1081:
1079:
1078:
1073:
1071:
1067:
1066:
1062:
1061:
1059:
1042:
1037:
1035:
1018:
1013:
1011:
994:
989:
987:
970:
963:
961:
960:
948:
940:
935:
896:
894:
893:
888:
883:
882:
864:
863:
851:
850:
810:
808:
807:
802:
797:
793:
792:
790:
789:
788:
763:
762:
753:
750:
745:
730:
728:
727:
715:
707:
702:
632:
630:
629:
624:
612:
610:
609:
604:
583:
581:
580:
575:
554:
552:
551:
546:
526:
524:
523:
518:
496:
494:
493:
488:
467:
465:
464:
459:
438:
436:
435:
430:
412:
410:
409:
404:
402:
398:
397:
395:
381:
367:
365:
363:
362:
350:
342:
337:
299:
291:
283:
275:
263:
251:
238:spigot algorithm
232:
230:
229:
224:
222:
218:
217:
213:
212:
210:
193:
188:
186:
169:
164:
162:
145:
140:
138:
121:
114:
112:
111:
99:
91:
86:
42:
21:
5106:
5105:
5101:
5100:
5099:
5097:
5096:
5095:
5076:
5075:
5066:David H. Bailey
5057:
5055:
5052:
5047:
5024:
5017:math.CA/9803067
5004:
5002:Further reading
4999:
4998:
4994:math.CA/9803067
4985:
4981:
4964:
4962:
4959:
4954:
4953:
4949:
4909:
4908:
4904:
4885:
4884:
4883:
4879:
4869:
4867:
4856:"PiHex Credits"
4854:
4853:
4849:
4830:
4829:
4828:
4824:
4814:
4812:
4809:
4804:
4803:
4799:
4791:
4787:
4739:
4738:
4731:
4726:
4706:
4701:
4667:
4666:
4647:
4646:
4619:
4614:
4613:
4586:
4581:
4580:
4553:
4548:
4547:
4515:
4514:
4486:
4485:
4457:
4456:
4432:
4427:
4426:
4405:
4400:
4399:
4392:
4390:Generalizations
4383:
4342:
4341:
4333:
4323:Fabrice Bellard
4306:
4303:
4299:
4255:
4242:
4229:
4213:
4205:
4204:
4197:
4173:Notice how the
4141:
4125:
4077:
4035:
4034:
4006:
4005:
3956:
3940:
3892:
3876:
3834:
3818:
3790:
3789:
3726:
3722:
3721:
3651:
3647:
3646:
3582:
3578:
3577:
3546:
3545:
3474:
3470:
3469:
3405:
3401:
3400:
3336:
3332:
3331:
3264:
3260:
3259:
3219:
3218:
3204:
3175:
3174:
3149:
3148:
3145:
3141:
3133:
3079:
3063:
3047:
3043:
3017:
3013:
3007:
2995:
2989:
2985:
2953:
2952:
2932:
2931:
2824:
2823:
2797:
2773:
2749:
2725:
2719:
2715:
2703:
2697:
2693:
2665:
2653:
2652:
2637:
2634:
2630:
2573:
2566:
2547:
2546:
2514:
2498:
2494:
2475:
2468:
2464:
2442:
2441:
2415:
2401:
2397:
2376:
2370:
2366:
2351:
2345:
2341:
2283:
2274:
2260:
2254:
2250:
2220:
2219:
2197:
2196:
2172:
2171:
2147:
2146:
2122:
2121:
2096:
2071:
2070:
2021:
2020:
1978:
1962:
1961:
1937:
1936:
1870:
1869:
1847:
1838:
1826:
1820:
1816:
1763:
1762:
1727:
1726:
1707:
1700:
1679:
1672:
1651:
1644:
1623:
1616:
1591:
1573:
1572:
1557:
1556:
1490:
1489:
1467:
1458:
1446:
1440:
1436:
1383:
1382:
1347:
1346:
1329:
1310:
1289:
1251:
1226:
1225:
1215:
1090:
1089:
1046:
1022:
998:
974:
968:
964:
952:
946:
942:
910:
909:
874:
855:
842:
828:
827:
780:
764:
754:
719:
713:
709:
650:
649:
643:
641:Specializations
615:
614:
586:
585:
557:
556:
537:
536:
503:
502:
470:
469:
441:
440:
421:
420:
382:
368:
354:
348:
344:
312:
311:
297:
289:
281:
273:
261:
249:
197:
173:
149:
125:
119:
115:
103:
97:
93:
61:
60:
50:David H. Bailey
40:
28:
23:
22:
15:
12:
11:
5:
5104:
5102:
5094:
5093:
5088:
5078:
5077:
5074:
5073:
5063:
5045:
5035:
5023:
5022:External links
5020:
5019:
5018:
5003:
5000:
4997:
4996:
4979:
4947:
4902:
4877:
4847:
4822:
4797:
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4715:
4710:
4700:
4697:
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3860:
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3840:
3837:
3831:
3828:
3825:
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3813:
3808:
3805:
3802:
3798:
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3554:
3523:
3522:
3511:
3505:
3502:
3499:
3496:
3493:
3490:
3486:
3481:
3477:
3473:
3468:
3461:
3456:
3453:
3450:
3446:
3442:
3436:
3433:
3430:
3427:
3424:
3421:
3417:
3412:
3408:
3404:
3399:
3392:
3387:
3384:
3381:
3377:
3373:
3367:
3364:
3361:
3358:
3355:
3352:
3348:
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3330:
3323:
3318:
3315:
3312:
3308:
3304:
3301:
3295:
3292:
3289:
3286:
3283:
3280:
3276:
3271:
3267:
3263:
3258:
3251:
3246:
3243:
3240:
3236:
3232:
3229:
3226:
3188:
3185:
3182:
3162:
3159:
3156:
3144:
3138:
3130:
3129:
3118:
3114:
3109:
3103:
3100:
3097:
3094:
3091:
3086:
3082:
3078:
3075:
3070:
3066:
3062:
3059:
3054:
3050:
3046:
3041:
3038:
3035:
3032:
3029:
3024:
3020:
3016:
3010:
3002:
2998:
2994:
2988:
2982:
2977:
2974:
2971:
2967:
2963:
2960:
2946:
2945:
2930:
2925:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2840:
2835:
2832:
2829:
2827:
2825:
2821:
2816:
2809:
2806:
2803:
2800:
2796:
2791:
2785:
2782:
2779:
2776:
2772:
2767:
2761:
2758:
2755:
2752:
2748:
2743:
2737:
2734:
2731:
2728:
2724:
2718:
2710:
2706:
2702:
2696:
2690:
2685:
2682:
2679:
2675:
2671:
2668:
2666:
2664:
2661:
2660:
2633:
2627:
2619:PSLQ algorithm
2565:
2562:
2561:
2560:
2545:
2541:
2536:
2532:
2529:
2524:
2521:
2517:
2513:
2510:
2507:
2504:
2501:
2497:
2493:
2490:
2487:
2482:
2478:
2474:
2471:
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2463:
2458:
2455:
2450:
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2445:
2443:
2439:
2434:
2427:
2424:
2421:
2418:
2411:
2408:
2404:
2400:
2394:
2388:
2385:
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2379:
2375:
2369:
2361:
2358:
2354:
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2344:
2338:
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2327:
2323:
2317:
2314:
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2286:
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2277:
2267:
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2247:
2242:
2239:
2236:
2232:
2228:
2225:
2223:
2221:
2218:
2215:
2209:
2204:
2200:
2195:
2190:
2184:
2179:
2175:
2170:
2165:
2159:
2154:
2150:
2145:
2140:
2134:
2129:
2125:
2120:
2115:
2110:
2107:
2102:
2099:
2097:
2093:
2090:
2085:
2082:
2079:
2078:
2052:
2051:
2036:
2033:
2028:
2024:
2019:
2012:
2007:
2004:
2001:
1997:
1993:
1987:
1984:
1981:
1977:
1972:
1969:
1951:
1950:
1935:
1930:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1896:
1891:
1886:
1883:
1878:
1875:
1873:
1871:
1867:
1862:
1856:
1853:
1850:
1846:
1841:
1833:
1829:
1825:
1819:
1813:
1808:
1805:
1802:
1798:
1792:
1789:
1784:
1778:
1775:
1770:
1766:
1761:
1754:
1749:
1746:
1743:
1739:
1735:
1732:
1730:
1728:
1725:
1722:
1714:
1710:
1706:
1703:
1699:
1694:
1686:
1682:
1678:
1675:
1671:
1666:
1658:
1654:
1650:
1647:
1643:
1638:
1630:
1626:
1622:
1619:
1615:
1610:
1605:
1602:
1597:
1594:
1592:
1590:
1587:
1584:
1581:
1580:
1570:
1555:
1550:
1545:
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1539:
1536:
1533:
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1524:
1521:
1516:
1511:
1506:
1503:
1498:
1495:
1493:
1491:
1487:
1482:
1476:
1473:
1470:
1466:
1461:
1453:
1449:
1445:
1439:
1433:
1428:
1425:
1422:
1418:
1412:
1409:
1404:
1398:
1395:
1390:
1386:
1381:
1374:
1369:
1366:
1363:
1359:
1355:
1352:
1350:
1348:
1345:
1342:
1336:
1332:
1328:
1323:
1317:
1313:
1309:
1304:
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1283:
1278:
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1234:
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1116:
1113:
1108:
1103:
1100:
1097:
1083:
1082:
1070:
1065:
1058:
1055:
1052:
1049:
1045:
1040:
1034:
1031:
1028:
1025:
1021:
1016:
1010:
1007:
1004:
1001:
997:
992:
986:
983:
980:
977:
973:
967:
959:
955:
951:
945:
939:
934:
931:
928:
924:
920:
917:
886:
881:
877:
873:
870:
867:
862:
858:
854:
849:
845:
841:
838:
835:
812:
811:
800:
796:
787:
783:
779:
776:
773:
770:
767:
761:
757:
749:
744:
741:
738:
734:
726:
722:
718:
712:
706:
701:
698:
695:
691:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
642:
639:
635:experimentally
622:
602:
599:
596:
593:
573:
570:
567:
564:
544:
527:is an integer
516:
513:
510:
486:
483:
480:
477:
457:
454:
451:
448:
428:
414:
413:
401:
394:
391:
388:
385:
380:
377:
374:
371:
361:
357:
353:
347:
341:
336:
333:
330:
326:
322:
319:
234:
233:
221:
216:
209:
206:
203:
200:
196:
191:
185:
182:
179:
176:
172:
167:
161:
158:
155:
152:
148:
143:
137:
134:
131:
128:
124:
118:
110:
106:
102:
96:
90:
85:
82:
79:
75:
71:
68:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5103:
5092:
5089:
5087:
5086:Pi algorithms
5084:
5083:
5081:
5071:
5067:
5064:
5051:
5046:
5043:
5039:
5036:
5033:
5029:
5026:
5025:
5021:
5016:
5015:
5010:
5006:
5005:
5001:
4993:
4989:
4983:
4980:
4976:
4974:
4958:
4951:
4948:
4943:
4939:
4935:
4931:
4927:
4923:
4919:
4915:
4914:
4906:
4903:
4897:
4896:
4891:
4890:"BBP Formula"
4888:
4881:
4878:
4865:
4861:
4857:
4851:
4848:
4842:
4841:
4836:
4833:
4826:
4823:
4808:
4801:
4798:
4794:
4789:
4786:
4781:
4777:
4772:
4767:
4762:
4757:
4753:
4749:
4748:
4743:
4736:
4734:
4730:
4723:
4719:
4716:
4714:
4711:
4709:
4703:
4702:
4698:
4696:
4694:
4678:
4675:
4672:
4652:
4632:
4629:
4624:
4620:
4599:
4596:
4591:
4587:
4566:
4563:
4558:
4554:
4545:
4526:
4520:
4497:
4491:
4468:
4462:
4455:
4437:
4433:
4410:
4406:
4397:
4389:
4387:
4381:
4362:
4359:
4356:
4353:
4347:
4339:
4330:
4328:
4324:
4320:
4316:
4312:
4297:
4295:
4292:
4288:
4283:
4265:
4260:
4252:
4247:
4239:
4234:
4226:
4223:
4218:
4210:
4203:
4202:
4201:
4194:
4192:
4188:
4184:
4180:
4176:
4157:
4151:
4148:
4145:
4142:
4136:
4133:
4130:
4126:
4113:
4110:
4107:
4104:
4101:
4097:
4093:
4087:
4084:
4081:
4078:
4070:
4067:
4064:
4061:
4046:
4043:
4040:
4036:
4027:
4022:
4019:
4016:
4012:
4004:
4003:
4002:
4000:
3996:
3992:
3972:
3966:
3963:
3960:
3957:
3951:
3948:
3945:
3941:
3928:
3925:
3922:
3919:
3916:
3912:
3908:
3902:
3899:
3896:
3893:
3887:
3884:
3881:
3877:
3869:
3864:
3861:
3858:
3854:
3850:
3844:
3841:
3838:
3835:
3829:
3826:
3823:
3819:
3806:
3803:
3800:
3796:
3788:
3787:
3786:
3784:
3780:
3761:
3752:
3749:
3746:
3743:
3736:
3731:
3727:
3723:
3718:
3706:
3703:
3700:
3697:
3694:
3690:
3686:
3677:
3674:
3671:
3668:
3661:
3656:
3652:
3648:
3643:
3636:
3631:
3628:
3625:
3621:
3617:
3608:
3605:
3602:
3599:
3592:
3587:
3583:
3579:
3574:
3562:
3559:
3556:
3552:
3544:
3543:
3542:
3540:
3536:
3532:
3528:
3509:
3500:
3497:
3494:
3491:
3484:
3479:
3475:
3471:
3466:
3454:
3451:
3448:
3444:
3440:
3431:
3428:
3425:
3422:
3415:
3410:
3406:
3402:
3397:
3385:
3382:
3379:
3375:
3371:
3362:
3359:
3356:
3353:
3346:
3341:
3337:
3333:
3328:
3316:
3313:
3310:
3306:
3302:
3299:
3290:
3287:
3284:
3281:
3274:
3269:
3265:
3261:
3256:
3244:
3241:
3238:
3234:
3230:
3227:
3224:
3217:
3216:
3215:
3212:
3210:
3202:
3186:
3183:
3180:
3160:
3157:
3154:
3139:
3137:
3116:
3112:
3107:
3101:
3098:
3095:
3092:
3089:
3084:
3080:
3076:
3073:
3068:
3064:
3060:
3057:
3052:
3048:
3044:
3039:
3036:
3033:
3030:
3027:
3022:
3018:
3014:
3008:
3000:
2996:
2992:
2986:
2975:
2972:
2969:
2965:
2961:
2958:
2951:
2950:
2949:
2928:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2858:
2855:
2852:
2849:
2846:
2843:
2833:
2830:
2828:
2819:
2814:
2807:
2804:
2801:
2798:
2794:
2789:
2783:
2780:
2777:
2774:
2770:
2765:
2759:
2756:
2753:
2750:
2746:
2741:
2735:
2732:
2729:
2726:
2722:
2716:
2708:
2704:
2700:
2694:
2683:
2680:
2677:
2673:
2669:
2667:
2662:
2651:
2650:
2649:
2647:
2643:
2628:
2626:
2624:
2620:
2616:
2612:
2608:
2604:
2600:
2595:
2591:
2587:
2583:
2579:
2571:
2563:
2543:
2539:
2534:
2530:
2527:
2522:
2519:
2515:
2511:
2508:
2505:
2502:
2499:
2495:
2491:
2488:
2485:
2480:
2476:
2472:
2469:
2465:
2461:
2456:
2453:
2448:
2446:
2437:
2432:
2425:
2422:
2419:
2416:
2409:
2406:
2402:
2398:
2392:
2386:
2383:
2380:
2377:
2373:
2367:
2359:
2356:
2352:
2348:
2342:
2331:
2328:
2325:
2321:
2315:
2312:
2307:
2303:
2297:
2291:
2287:
2284:
2278:
2275:
2265:
2261:
2257:
2251:
2240:
2237:
2234:
2230:
2226:
2224:
2216:
2213:
2207:
2202:
2198:
2193:
2188:
2182:
2177:
2173:
2168:
2163:
2157:
2152:
2148:
2143:
2138:
2132:
2127:
2123:
2118:
2113:
2108:
2105:
2100:
2098:
2091:
2088:
2083:
2080:
2069:
2068:
2067:
2065:
2061:
2057:
2034:
2031:
2026:
2022:
2017:
2005:
2002:
1999:
1995:
1991:
1985:
1982:
1979:
1975:
1970:
1967:
1960:
1959:
1958:
1956:
1933:
1920:
1914:
1911:
1908:
1905:
1902:
1899:
1889:
1884:
1881:
1876:
1874:
1865:
1860:
1854:
1851:
1848:
1844:
1839:
1831:
1827:
1823:
1817:
1806:
1803:
1800:
1796:
1790:
1787:
1782:
1776:
1773:
1768:
1764:
1759:
1747:
1744:
1741:
1737:
1733:
1731:
1723:
1720:
1712:
1708:
1704:
1701:
1697:
1692:
1684:
1680:
1676:
1673:
1669:
1664:
1656:
1652:
1648:
1645:
1641:
1636:
1628:
1624:
1620:
1617:
1613:
1608:
1603:
1600:
1595:
1593:
1588:
1585:
1582:
1571:
1553:
1540:
1534:
1531:
1528:
1525:
1522:
1519:
1509:
1504:
1501:
1496:
1494:
1485:
1480:
1474:
1471:
1468:
1464:
1459:
1451:
1447:
1443:
1437:
1426:
1423:
1420:
1416:
1410:
1407:
1402:
1396:
1393:
1388:
1384:
1379:
1367:
1364:
1361:
1357:
1353:
1351:
1343:
1340:
1334:
1330:
1326:
1321:
1315:
1311:
1307:
1302:
1296:
1290:
1286:
1281:
1276:
1273:
1268:
1263:
1260:
1255:
1253:
1246:
1243:
1238:
1235:
1224:
1223:
1222:
1220:
1212:
1196:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1126:
1123:
1120:
1117:
1114:
1111:
1101:
1098:
1095:
1088:
1087:
1086:
1068:
1063:
1056:
1053:
1050:
1047:
1043:
1038:
1032:
1029:
1026:
1023:
1019:
1014:
1008:
1005:
1002:
999:
995:
990:
984:
981:
978:
975:
971:
965:
957:
953:
949:
943:
932:
929:
926:
922:
918:
915:
908:
907:
906:
904:
900:
879:
875:
871:
868:
865:
860:
856:
852:
847:
843:
836:
833:
825:
821:
817:
798:
794:
785:
777:
774:
771:
768:
759:
755:
747:
742:
739:
736:
732:
724:
720:
716:
710:
699:
696:
693:
689:
685:
679:
676:
673:
670:
667:
664:
661:
655:
648:
647:
646:
640:
638:
636:
620:
597:
591:
568:
562:
542:
534:
530:
514:
511:
508:
500:
481:
475:
452:
446:
426:
419:
399:
389:
383:
375:
369:
359:
355:
351:
345:
334:
331:
328:
324:
320:
317:
310:
309:
308:
305:
303:
295:
287:
279:
271:
267:
259:
255:
247:
243:
239:
219:
214:
207:
204:
201:
198:
194:
189:
183:
180:
177:
174:
170:
165:
159:
156:
153:
150:
146:
141:
135:
132:
129:
126:
122:
116:
108:
104:
100:
94:
83:
80:
77:
73:
69:
66:
59:
58:
57:
55:
54:Peter Borwein
51:
47:
46:Simon Plouffe
43:
37:
33:
19:
5056:. Retrieved
5012:
4991:
4982:
4972:
4970:
4963:. Retrieved
4950:
4920:(1): 50–57.
4917:
4911:
4905:
4893:
4880:
4868:. Retrieved
4859:
4850:
4838:
4825:
4813:. Retrieved
4800:
4788:
4751:
4745:
4393:
4379:
4338:linearithmic
4331:
4318:
4314:
4310:
4304:
4284:
4280:
4195:
4190:
4178:
4172:
3998:
3994:
3990:
3987:
3782:
3778:
3776:
3538:
3526:
3524:
3213:
3146:
3131:
2947:
2645:
2635:
2622:
2606:
2602:
2598:
2593:
2589:
2585:
2581:
2577:
2569:
2567:
2063:
2059:
2053:
1954:
1952:
1218:
1216:
1084:
902:
823:
819:
815:
813:
644:
532:
415:
306:
301:
296:th digit of
293:
280:th digit of
277:
269:
268:compute the
265:
258:binary digit
253:
241:
235:
35:
31:
29:
3537:across the
3201:hexadecimal
3173:)-th (with
2613:(typically
499:polynomials
36:BBP formula
5080:Categories
5058:2019-03-31
4965:17 January
4815:4 November
4724:References
4513:, (where
2648:function:
2568:Using the
5011:, (1998)
4990:, (1998)
4895:MathWorld
4840:MathWorld
4676:
4653:π
4630:
4597:
4564:
4521:ζ
4492:ζ
4463:ζ
4434:π
4407:π
4360:
4313:th digit
4257:Σ
4253:−
4244:Σ
4240:−
4231:Σ
4224:−
4215:Σ
4134:−
4119:∞
4098:∑
4044:−
4013:∑
3949:−
3934:∞
3913:∑
3885:−
3855:∑
3827:−
3812:∞
3797:∑
3712:∞
3691:∑
3622:∑
3568:∞
3553:∑
3541:th term:
3460:∞
3445:∑
3441:−
3391:∞
3376:∑
3372:−
3322:∞
3307:∑
3300:−
3250:∞
3235:∑
3225:π
3203:digit of
3184:≥
2981:∞
2966:∑
2959:π
2901:−
2892:−
2883:−
2790:−
2766:−
2742:−
2689:∞
2674:∑
2663:π
2520:−
2512:−
2407:−
2399:−
2337:∞
2322:∑
2288:π
2279:
2246:∞
2231:∑
2217:⋯
2164:−
2114:−
2084:
2032:⋅
2011:∞
1996:∑
1983:−
1971:
1812:∞
1797:∑
1774:⋅
1753:∞
1738:∑
1724:⋯
1705:⋅
1677:⋅
1649:⋅
1621:⋅
1586:
1432:∞
1417:∑
1394:⋅
1373:∞
1358:∑
1344:⋯
1239:
1169:−
1160:−
1151:−
1096:π
1039:−
1015:−
991:−
938:∞
923:∑
916:π
869:…
733:∑
705:∞
690:∑
543:α
512:≥
427:α
340:∞
325:∑
318:α
190:−
166:−
142:−
89:∞
74:∑
67:π
4942:14318695
4870:30 March
4864:Archived
4699:See also
3535:infinity
1957:> 1:
899:sequence
439:, where
304:digits.
4934:1439159
4780:1415794
4542:is the
4315:without
4291:carries
4175:modulus
2576:is for
246:base-16
4940:
4932:
4778:
4187:nested
2605:, and
2081:arctan
1294:
822:, and
814:where
613:, and
5053:(PDF)
5014:arXiv
4992:arXiv
4960:(PDF)
4938:S2CID
4810:(PDF)
3783:(n+1)
3779:n = 0
897:is a
286:PiHex
4967:2013
4872:2018
4817:2020
4665:and
3061:1024
2642:PSLQ
529:base
497:are
468:and
30:The
5068:, "
5040:, "
5030:, "
4922:doi
4766:hdl
4756:doi
4673:log
4621:log
4588:log
4555:log
4546:),
4357:log
4053:mod
3533:to
3531:sum
3093:194
3077:712
3045:512
3031:151
3015:120
2617:'s
2276:sin
1335:000
1331:500
1316:000
1297:000
1277:200
266:not
260:of
256:th
244:th
5082::
4969:.
4936:.
4930:MR
4928:.
4918:19
4916:.
4892:.
4858:.
4837:.
4776:MR
4774:.
4764:.
4752:66
4750:.
4744:.
4732:^
4695:.
4612:,
4579:,
4484:,
4452:,
4425:,
4398:,
4200::
4127:16
4037:16
3942:16
3878:16
3820:16
3728:16
3653:16
3584:16
3476:16
3407:16
3338:16
3266:16
3199:)
3136:.
3102:15
3040:47
2997:16
2850:16
2705:16
2601:,
2050:.)
1968:ln
1583:ln
1526:10
1505:10
1448:10
1411:10
1385:10
1312:40
1264:10
1244:10
1236:ln
1118:16
954:16
818:,
637:.
584:,
254:4n
105:16
52:,
5061:.
4975:.
4973:d
4944:.
4924::
4898:.
4874:.
4843:.
4819:.
4795:.
4782:.
4768::
4758::
4707:π
4679:2
4633:2
4625:5
4600:2
4592:4
4567:2
4559:3
4530:)
4527:x
4524:(
4501:)
4498:5
4495:(
4472:)
4469:3
4466:(
4438:4
4411:3
4384:π
4380:n
4366:)
4363:n
4354:n
4351:(
4348:O
4340:(
4334:π
4319:n
4311:n
4307:π
4300:π
4266:.
4261:4
4248:3
4235:2
4227:2
4219:1
4211:4
4198:π
4191:x
4179:k
4158:.
4152:1
4149:+
4146:k
4143:8
4137:k
4131:n
4114:1
4111:+
4108:n
4105:=
4102:k
4094:+
4088:1
4085:+
4082:k
4079:8
4074:)
4071:1
4068:+
4065:k
4062:8
4057:(
4047:k
4041:n
4028:n
4023:0
4020:=
4017:k
3999:k
3995:n
3991:k
3973:.
3967:1
3964:+
3961:k
3958:8
3952:k
3946:n
3929:1
3926:+
3923:n
3920:=
3917:k
3909:+
3903:1
3900:+
3897:k
3894:8
3888:k
3882:n
3870:n
3865:0
3862:=
3859:k
3851:=
3845:1
3842:+
3839:k
3836:8
3830:k
3824:n
3807:0
3804:=
3801:k
3762:.
3756:)
3753:1
3750:+
3747:k
3744:8
3741:(
3737:)
3732:k
3724:(
3719:1
3707:1
3704:+
3701:n
3698:=
3695:k
3687:+
3681:)
3678:1
3675:+
3672:k
3669:8
3666:(
3662:)
3657:k
3649:(
3644:1
3637:n
3632:0
3629:=
3626:k
3618:=
3612:)
3609:1
3606:+
3603:k
3600:8
3597:(
3593:)
3588:k
3580:(
3575:1
3563:0
3560:=
3557:k
3539:n
3527:n
3510:.
3504:)
3501:6
3498:+
3495:k
3492:8
3489:(
3485:)
3480:k
3472:(
3467:1
3455:0
3452:=
3449:k
3435:)
3432:5
3429:+
3426:k
3423:8
3420:(
3416:)
3411:k
3403:(
3398:1
3386:0
3383:=
3380:k
3366:)
3363:4
3360:+
3357:k
3354:8
3351:(
3347:)
3342:k
3334:(
3329:1
3317:0
3314:=
3311:k
3303:2
3294:)
3291:1
3288:+
3285:k
3282:8
3279:(
3275:)
3270:k
3262:(
3257:1
3245:0
3242:=
3239:k
3231:4
3228:=
3205:π
3187:0
3181:n
3161:1
3158:+
3155:n
3142:π
3134:π
3117:.
3113:]
3108:)
3099:+
3096:k
3090:+
3085:2
3081:k
3074:+
3069:3
3065:k
3058:+
3053:4
3049:k
3037:+
3034:k
3028:+
3023:2
3019:k
3009:(
3001:k
2993:1
2987:[
2976:0
2973:=
2970:k
2962:=
2929:,
2924:)
2919:)
2916:0
2913:,
2910:0
2907:,
2904:1
2898:,
2895:1
2889:,
2886:2
2880:,
2877:0
2874:,
2871:0
2868:,
2865:4
2862:(
2859:,
2856:8
2853:,
2847:,
2844:1
2839:(
2834:P
2831:=
2820:]
2815:)
2808:6
2805:+
2802:k
2799:8
2795:1
2784:5
2781:+
2778:k
2775:8
2771:1
2760:4
2757:+
2754:k
2751:8
2747:2
2736:1
2733:+
2730:k
2727:8
2723:4
2717:(
2709:k
2701:1
2695:[
2684:0
2681:=
2678:k
2670:=
2646:P
2638:π
2631:π
2623:A
2607:m
2603:b
2599:s
2594:A
2590:m
2586:b
2582:m
2578:s
2574:π
2570:P
2544:.
2540:)
2535:)
2531:0
2528:,
2523:2
2516:b
2509:,
2506:0
2503:,
2500:1
2496:(
2492:,
2489:4
2486:,
2481:4
2477:b
2473:,
2470:1
2466:(
2462:P
2457:b
2454:1
2449:=
2438:]
2433:)
2426:3
2423:+
2420:k
2417:4
2410:2
2403:b
2393:+
2387:1
2384:+
2381:k
2378:4
2374:1
2368:(
2360:k
2357:4
2353:b
2349:1
2343:[
2332:0
2329:=
2326:k
2316:b
2313:1
2308:=
2304:]
2298:k
2292:2
2285:k
2266:k
2262:b
2258:1
2252:[
2241:1
2238:=
2235:k
2227:=
2214:+
2208:9
2203:9
2199:b
2194:1
2189:+
2183:7
2178:7
2174:b
2169:1
2158:5
2153:5
2149:b
2144:1
2139:+
2133:3
2128:3
2124:b
2119:1
2109:b
2106:1
2101:=
2092:b
2089:1
2064:b
2060:P
2035:k
2027:k
2023:a
2018:1
2006:1
2003:=
2000:k
1992:=
1986:1
1980:a
1976:a
1955:a
1934:.
1929:)
1924:)
1921:1
1918:(
1915:,
1912:1
1909:,
1906:2
1903:,
1900:1
1895:(
1890:P
1885:2
1882:1
1877:=
1866:]
1861:)
1855:1
1852:+
1849:k
1845:1
1840:(
1832:k
1828:2
1824:1
1818:[
1807:0
1804:=
1801:k
1791:2
1788:1
1783:=
1777:k
1769:k
1765:2
1760:1
1748:1
1745:=
1742:k
1734:=
1721:+
1713:5
1709:2
1702:5
1698:1
1693:+
1685:4
1681:2
1674:4
1670:1
1665:+
1657:3
1653:2
1646:3
1642:1
1637:+
1629:2
1625:2
1618:2
1614:1
1609:+
1604:2
1601:1
1596:=
1589:2
1554:,
1549:)
1544:)
1541:1
1538:(
1535:,
1532:1
1529:,
1523:,
1520:1
1515:(
1510:P
1502:1
1497:=
1486:]
1481:)
1475:1
1472:+
1469:k
1465:1
1460:(
1452:k
1444:1
1438:[
1427:0
1424:=
1421:k
1408:1
1403:=
1397:k
1389:k
1380:1
1368:1
1365:=
1362:k
1354:=
1341:+
1327:1
1322:+
1308:1
1303:+
1291:3
1287:1
1282:+
1274:1
1269:+
1261:1
1256:=
1247:9
1219:P
1197:.
1192:)
1187:)
1184:0
1181:,
1178:0
1175:,
1172:1
1166:,
1163:1
1157:,
1154:2
1148:,
1145:0
1142:,
1139:0
1136:,
1133:4
1130:(
1127:,
1124:8
1121:,
1115:,
1112:1
1107:(
1102:P
1099:=
1069:]
1064:)
1057:6
1054:+
1051:k
1048:8
1044:1
1033:5
1030:+
1027:k
1024:8
1020:1
1009:4
1006:+
1003:k
1000:8
996:2
985:1
982:+
979:k
976:8
972:4
966:(
958:k
950:1
944:[
933:0
930:=
927:k
919:=
903:P
885:)
880:m
876:a
872:,
866:,
861:2
857:a
853:,
848:1
844:a
840:(
837:=
834:A
824:m
820:b
816:s
799:,
795:]
786:s
782:)
778:j
775:+
772:k
769:m
766:(
760:j
756:a
748:m
743:1
740:=
737:j
725:k
721:b
717:1
711:[
700:0
697:=
694:k
686:=
683:)
680:A
677:,
674:m
671:,
668:b
665:,
662:s
659:(
656:P
621:b
601:)
598:k
595:(
592:q
572:)
569:k
566:(
563:p
515:2
509:b
485:)
482:k
479:(
476:q
456:)
453:k
450:(
447:p
400:]
393:)
390:k
387:(
384:q
379:)
376:k
373:(
370:p
360:k
356:b
352:1
346:[
335:0
332:=
329:k
321:=
302:n
298:π
294:n
290:π
282:π
278:n
274:π
270:n
262:π
250:π
242:n
220:]
215:)
208:6
205:+
202:k
199:8
195:1
184:5
181:+
178:k
175:8
171:1
160:4
157:+
154:k
151:8
147:2
136:1
133:+
130:k
127:8
123:4
117:(
109:k
101:1
95:[
84:0
81:=
78:k
70:=
41:π
34:(
20:)
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