M1*arctan(1/A1)+M2*arctan(1/A2)+...+Mj*arctan(1/Aj) == k*Pi/4the left hand side is abbreviated as

M1[A1]+M2[A2]+...+Mj[Aj]The term of least convergence is listed first. Relations of n arctan terms are in one file. The files are ordered according to the arguments, the "best" relation is first. When the first arguments coincide the next is used for ordering. An example (6-term relations):

+322[577] +76[682] +139[1393] +156[12943] +132[32807] +44[1049433] == 1 * Pi/4 +122[319] +61[378] +115[557] +29[1068] +22[3458] +44[27493] == 1 * Pi/4 +100[319] +127[378] +71[557] -15[1068] +66[2943] +44[478707] == 1 * Pi/4 +337[307] -193[463] +151[4193] +305[4246] -122[39307] -83[390112] == 1 * Pi/4 +183[268] +32[682] +95[1568] +44[4662] -166[12943] -51[32807] == 1 * Pi/4 +183[268] +32[682] +95[1483] -7[9932] -122[12943] +51[29718] == 1 * Pi/4 +29[268] +269[463] +154[2059] +122[2943] -186[9193] +71[390112] == 1 * Pi/4

Each relation is followed by a list of primes of the form 4*k+1. These are obtained by factoring Ai^2+1 for each (inverse) argument Ai. An example (a 5-term relation):

+88[192] +39[239] +100[515] -32[1068] -56[173932] == 1 * Pi/4 {5, 13, 73, 101}We have

192^2+1 == 36865 == 5 73 101 239^2+1 == 57122 == 2 13 13 13 13 515^2+1 == 265226 == 2 13 101 101 1068^2+1 == 1140625 == 5 5 5 5 5 5 73 173932^2+1 == 30252340625 == 5 5 5 5 5 13 73 101 101

The search is described in the fxtbook. Here are the slides of my talk "Search for the best arctan relation" given 2006 in Berlin (gzip compressed): dvi (8kB), ps (75kB), or pdf (80kB).

All relations were computed April-2006. Some of them improve on my April-1993 computation. The files near the bottom of the hfloat page contain the (now obsolete) 1993 data.

- The few "best" relations for each number N of terms: best-arctan-relations.txt
- The single best relation for each N: best-short.txt
- 2-term relations: arctan-2term.txt
- 3-term relations: arctan-3term.txt
- 4-term relations: arctan-4term.txt
- 5-term relations: arctan-5term.txt
- 6-term relations: arctan-6term.txt
- 7-term relations: arctan-7term.txt
- 8-term relations: arctan-8term.txt
- 9-term relations: arctan-9term.txt
- 10-term relations: arctan-10term.txt
- 11-term relations: arctan-11term.txt
- 12-term relations: arctan-12term.txt
- 13-term relations: arctan-13term.txt
- 14-term relations: arctan-14term.txt
- 15-term relations: arctan-15term.txt
- 16-term relations: arctan-16term.txt
- 17-term relations: arctan-17term.txt
- 18-term relations: arctan-18term.txt
- 19-term relations: arctan-19term.txt
- 20-term relations: arctan-20term.txt
- 21-term relations: arctan-21term.txt
- 22-term relations: arctan-22term.txt
- 23-term relations: arctan-23term.txt
- 24-term relations: arctan-24term.txt
- 25-term relations: arctan-25term.txt
- 26-term relations: arctan-26term.txt
- 27-term relations: arctan-27term.txt

Note added 2010: Amrik Singh Nimbran communicated (23-July-2010) a 25-term identity (arctan-25term-nimbran.gp) which is 'better' than my best 25-term relation: the first argument is 218,123,852,367 (my identity has 160,422,360,532). The prime 941 involved shows that my search could not have found this relation.

He also gave (27-October-2010) a 20-term relation (arctan-20term-nimbran.gp) with first argument 3,739,944,528 (my identity has 2,674,664,693). The primes 977 and 1409 were not used in my 2006 search.

Also (2-June-2011) a 24 term relation (arctan-24term-nimbran.gp) with first argument 121,409,547,033 (improving on my 102,416,588,812). The prime 941 was not included in my search.

n-terms min-arg 2 5 Machin (1706) 3 18 Gauss (YY?) 4 57 Stormer (1896) 5 192 JJ (1993), prev: Stormer (1896) 172 6 577 JJ (1993) 7 2,852 JJ (1993) 8 5,357 JJ (2006), prev: JJ (1993) 4,246 9 34,208 JJ (2006), prev: JJ (1993) 12,943, prev: Gauss (Y?) 5,257 10 54,193 JJ (2006), prev: JJ (1993) 51,387 11 390,112 JJ (1993) 12 1,049,433 JJ (2006), prev: JJ (1993) 683,982 13 3,449,051 JJ (2006), prev: JJ (1993) 1,984,933 14 6,826,318 JJ (2006) 15 20,942,043 HCL (1997), prev: MRW (1997) 18.975,991 16 53,141,564 JJ (2006) 17 201,229,582 JJ (2006) 18 299,252,491 JJ (2006) 19 778,401,733 JJ (2006) 20 2,674,664,693 JJ (2006) [Note: superseded by Nimbran: 3,739,944,528 (2010)] 21 5,513,160,193 JJ (2006) 22 17,249,711,432 JJ (2006), prev: 16,077,395,443 MRW (27-Jan-2003) 23 58,482,499,557 JJ (2006) 24 102,416,588,812 JJ (2006) [Note: superseded by Nimbran: 121,409,547,033 (2011)] 25 160,422,360,532 JJ (2006) [Note: superseded by Nimbran: 218,123,852,367 (2010)] 26 392,943,720,343 JJ (2006) 27 970,522,492,753 JJ (2006) MRW := Michael Roby Wetherfield HCL := Hwang Chien-lih JJ := Joerg ArndtI am indebted to Michael Roby Wetherfield who supplied a list of arguments X (so that X^2+1 is 761-smooth) beyond the range (10^14) of my exhaustive search. His web site is here.

Your feedback is appreciated.

jj (Jörg Arndt)

Last modified 2018-October-11 (18:29)

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