Arctan relations for Pi

Description of the files

In a relation

 M1*arctan(1/A1)+M2*arctan(1/A2)+...+Mj*arctan(1/Aj) == k*Pi/4

the 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 relations

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.

The state of art

 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 Arndt

I 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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