Formulas of the Form

k \frac{π}{4} = N ; ∑ ; i=1 m_{i} arctan (\frac{1}{x_{i}})

By Jörg Arndt <arndt@jjj.de> This document is online at http://www.jjj.de/

28-August-1996

the 1-term formula:

\frac{π}{4} = arctan(1)  (1)
{}

2-term formulae:

\frac{π}{4} = arctan (\frac{1}{2}) + arctan (\frac{1}{3})  (2)
{5} (Euler, 1706)
\frac{π}{4} = 2 arctan (\frac{1}{2}) − arctan (\frac{1}{7})  (3)
{5} (Herman, 1706)
\frac{π}{4} = 2 arctan (\frac{1}{3}) + arctan (\frac{1}{7})  (4)
{5} (Hutton, 1776)
\frac{π}{4} = 4 arctan (\frac{1}{5}) − arctan (\frac{1}{239})  (5)
{13} (Machin, 1706)

3-term formulas:

\frac{π}{4} = 2 arctan (\frac{1}{2}) − arctan (\frac{1}{9}) − arctan (\frac{1}{32})  (6)

{5,41}

\frac{π}{2} = 3 arctan (\frac{1}{2}) + 2 arctan (\frac{1}{13}) + arctan (\frac{1}{38})  (7)

{5,17}

\frac{π}{2} = 5 arctan (\frac{1}{3}) − 2 arctan (\frac{1}{68}) − arctan (\frac{1}{117})  (8)

{5,37}

\frac{π}{4} = 3 arctan (\frac{1}{4}) + arctan (\frac{1}{20}) + arctan (\frac{1}{1985})  (9)

{17,401}

\frac{π}{4} = 5 arctan (\frac{1}{7}) + 2 arctan (\frac{1}{23}) − 2 arctan (\frac{1}{182})  (10)

{5,53}

\frac{π}{4} = 5 arctan (\frac{1}{7}) + 2 arctan (\frac{1}{26}) − 2 arctan (\frac{1}{2057})  (11)

{5,677}

\frac{π}{4} = 5 arctan (\frac{1}{7}) + 2 arctan (\frac{1}{27}) + 2 arctan (\frac{1}{1068})  (12)

{5,73}

\frac{π}{4} = 5 arctan (\frac{1}{7}) + 2 arctan (\frac{1}{28}) + 2 arctan (\frac{1}{443})  (13)

{5,157}

\frac{π}{4} = 5 arctan (\frac{1}{7}) + 2 arctan (\frac{1}{43}) + 2 arctan (\frac{1}{68})  (14)

{5,37}

\frac{π}{4} = 5 arctan (\frac{1}{7}) + 4 arctan (\frac{1}{53}) + 2 arctan (\frac{1}{4443})  (15)

{5,181}

\frac{π}{4} = 6 arctan (\frac{1}{8}) + 2 arctan (\frac{1}{57}) + arctan (\frac{1}{239})  (16)

{5,13} (Størmer, 1896)

\frac{π}{4} = 8 arctan (\frac{1}{10}) − arctan (\frac{1}{239}) − 4 arctan (\frac{1}{515})  (17)

{13,101} (Klingstierna)

\frac{π}{4} = 12 arctan (\frac{1}{18}) + 8 arctan (\frac{1}{57}) − 5 arctan (\frac{1}{239})  (18)

{5,13} (Gauss)

4-term formulas:

\frac{π}{4} = 3 arctan (\frac{1}{7}) + 4 arctan (\frac{1}{13}) + 2 arctan (\frac{1}{55}) + 2 arctan (\frac{1}{123})  (19)

{ 5,17,89}

2 \frac{π}{4} = 11 arctan (\frac{1}{7}) + 4 arctan (\frac{1}{443}) + arctan (\frac{1}{1393}) + 2 arctan (\frac{1}{11018})  (20)

{5,157,197}

5 \frac{π}{4} = 44 arctan (\frac{1}{11}) − 9 arctan (\frac{1}{239}) − 16 arctan (\frac{1}{682}) − 12 arctan (\frac{1}{12943})  (21)

{5,13,61}

\frac{π}{4} = 12 arctan (\frac{1}{15}) − arctan (\frac{1}{239}) − 4 arctan (\frac{1}{580}) − 4 arctan (\frac{1}{1710})  (22)

{ 13,113,229}

\frac{π}{4} = 10 arctan (\frac{1}{17}) + 2 arctan (\frac{1}{38}) + 7 arctan (\frac{1}{41}) − 4 arctan (\frac{1}{157})  (23)

{5,17,29}

\frac{π}{4} = 12 arctan (\frac{1}{17}) + 8 arctan (\frac{1}{57}) − 5 arctan (\frac{1}{239}) − 12 arctan (\frac{1}{307})  (24)

{5,13,29}

\frac{π}{4} = 12 arctan (\frac{1}{18}) + 8 arctan (\frac{1}{73}) + 3 arctan (\frac{1}{239}) − 8 arctan (\frac{1}{2943})  (25)

{5,13,41}

\frac{π}{4} = 12 arctan (\frac{1}{18}) + 3 arctan (\frac{1}{70}) + 5 arctan (\frac{1}{99}) + 8 arctan (\frac{1}{307})  (26)

{5,13,29} (Bennet)

\frac{π}{4} = 12 arctan (\frac{1}{18}) + 8 arctan (\frac{1}{99}) + 3 arctan (\frac{1}{239}) + 8 arctan (\frac{1}{307})  (27)

{5,13,29} (D.H. Lehmer, 1938)

\frac{π}{4} = 12 arctan (\frac{1}{18}) + 8 arctan (\frac{1}{109}) + 11 arctan (\frac{1}{239}) − 8 arctan (\frac{1}{6826318})  (28)

{5,13,457}

3 \frac{π}{4} = 44 arctan (\frac{1}{18}) − 23 arctan (\frac{1}{239}) + 8 arctan (\frac{1}{682}) − 16 arctan (\frac{1}{12943})  (29)

{5,13,61}

\frac{π}{4} = 16 arctan (\frac{1}{21}) + 3 arctan (\frac{1}{239}) + 4 arctan (\frac{1}{421}) + 4 arctan (\frac{1}{1985})  (30)

{13,17,401}

\frac{π}{4} = 10 arctan (\frac{1}{22}) + 7 arctan (\frac{1}{41}) + 12 arctan (\frac{1}{75}) + 2 arctan (\frac{1}{4193})  (31)

{5,29,97} (Størmer, 1896)

\frac{π}{4} = 17 arctan (\frac{1}{23}) + 8 arctan (\frac{1}{182}) + 10 arctan (\frac{1}{5118}) + 5 arctan (\frac{1}{6072})  (32)

{5,53,373}

\frac{π}{4} = 17 arctan (\frac{1}{26}) + 5 arctan (\frac{1}{38}) + 3 arctan (\frac{1}{2057}) − 10 arctan (\frac{1}{8827})  (33)
{5,17,677}
\frac{π}{4} = 12 arctan (\frac{1}{28}) + 20 arctan (\frac{1}{57}) − 5 arctan (\frac{1}{239}) + 12 arctan (\frac{1}{443})  (34)
{ 5,13,157}
\frac{π}{4} = 22 arctan (\frac{1}{28}) + 2 arctan (\frac{1}{443}) − 5 arctan (\frac{1}{1393}) − 10 arctan (\frac{1}{11018})  (35)
{ 5,157,197} (Escott)
\frac{π}{4} = 24 arctan (\frac{1}{29}) − 4 arctan (\frac{1}{57}) + 7 arctan (\frac{1}{239}) − 12 arctan (\frac{1}{12238})  (36)
{5,13,421}
\frac{π}{4} = 12 arctan (\frac{1}{38}) + 20 arctan (\frac{1}{57}) + 7 arctan (\frac{1}{239}) + 24 arctan (\frac{1}{268})  (37)
{ 5,13,17} (Gauss)
\frac{π}{4} = 12 arctan (\frac{1}{49}) + 32 arctan (\frac{1}{57}) − 5 arctan (\frac{1}{239}) + 12 arctan (\frac{1}{110443})  (38)
{5,13,1201}
\frac{π}{4} = 24 arctan (\frac{1}{53}) + 20 arctan (\frac{1}{57}) − 5 arctan (\frac{1}{239}) + 12 arctan (\frac{1}{4443})  (39)
{5,13,281}
\frac{π}{4} = 20 arctan (\frac{1}{57}) + 24 arctan (\frac{1}{68}) + 12 arctan (\frac{1}{117}) − 5 arctan (\frac{1}{239})  (40)
{5,13,37}
\frac{π}{4} = 44 arctan (\frac{1}{57}) + 7 arctan (\frac{1}{239}) − 12 arctan (\frac{1}{682}) + 24 arctan (\frac{1}{12943})  (41)
{5,13,61} (Størmer, 1896)

5-term formulas:

\frac{π}{2} = 11 arctan (\frac{1}{7}) + 2 arctan (\frac{1}{99}) + arctan (\frac{1}{1393}) +  (42)
−2 arctan (\frac{1}{182}) − 2 arctan (\frac{1}{1264557})
{5,13,29,53,197}
\frac{π}{4} = 22 arctan (\frac{1}{29}) + 12 arctan (\frac{1}{392}) + 2 arctan (\frac{1}{1068}) −  (43)
−5 arctan (\frac{1}{1393}) − 10 arctan (\frac{1}{5502})
{5,73,197,421}
\frac{π}{4} = 22 arctan (\frac{1}{32}) + 17 arctan (\frac{1}{182}) + 7 arctan (\frac{1}{818}) −  (44)
−5 arctan (\frac{1}{1303}) − 12 arctan (\frac{1}{3014557})
{5,41,53,101}
\frac{π}{4} = 68 arctan (\frac{1}{99}) − 12 arctan (\frac{1}{117}) + 39 arctan (\frac{1}{239}) +  (45)
+20 arctan (\frac{1}{307}) − 24 arctan (\frac{1}{882})
{5,13,29,37}
\frac{π}{4} = 56 arctan (\frac{1}{99}) + 27 arctan (\frac{1}{239}) + 32 arctan (\frac{1}{307}) +  (46)
+12 arctan (\frac{1}{4193}) − 12 arctan (\frac{1}{39307})
{5,13,29,97}
\frac{π}{4} = 44 arctan (\frac{1}{109}) + 95 arctan (\frac{1}{239}) − 12 arctan (\frac{1}{682}) +  (47)
+24 arctan (\frac{1}{12943}) − 44 arctan (\frac{1}{6826318})
{5,13,61,457}
\frac{π}{4} = 88 arctan (\frac{1}{111}) + 7 arctan (\frac{1}{239}) − 44 arctan (\frac{1}{515}) +  (48)
+32 arctan (\frac{1}{682}) + 24 arctan (\frac{1}{12943})
{5,13,61,101}
\frac{π}{4} = 88 arctan (\frac{1}{172}) + 51 arctan (\frac{1}{239}) + 32 arctan (\frac{1}{682}) +  (49)
+44 arctan (\frac{1}{5357}) + 68 arctan (\frac{1}{12943})
{5,13,61,97} (Størmer, 1896)
\frac{π}{4} = 88 arctan (\frac{1}{192}) + 39 arctan (\frac{1}{239}) + 100 arctan (\frac{1}{515}) −  (50)
−32 arctan (\frac{1}{1068}) − 56 arctan (\frac{1}{173932})
{5,13,73,101}

134 \frac{π}{4} = 227 arctan (\frac{1}{2}) − 13 arctan (\frac{1}{682}) + 12 arctan (\frac{1}{2059}) +  (51)

+24 arctan (\frac{1}{2943}) − 6 arctan (\frac{1}{5357}) + 20 arctan (\frac{1}{12943})

{5,13,41,61,97}

3 \frac{π}{4} = 200 arctan (\frac{1}{99}) + 81 arctan (\frac{1}{239}) + 4 arctan (\frac{1}{4193}) −  (52)

−32 arctan (\frac{1}{12238}) − 4 arctan (\frac{1}{39307}) − 64 arctan (\frac{1}{58911})

{5,13,29,97,421}

\frac{π}{4} = 100 arctan (\frac{1}{216}) + 71 arctan (\frac{1}{265}) + 24 arctan (\frac{1}{1068}) +  (53)

+39 arctan (\frac{1}{2436}) + 44 arctan (\frac{1}{5507}) + 56 arctan (\frac{1}{6962})

{5,13,37,73,97}

\frac{π}{4} = 7 arctan (\frac{1}{239}) + 176 arctan (\frac{1}{293}) + 44 arctan (\frac{1}{616}) +  (54)

+32 arctan (\frac{1}{682}) + 132 arctan (\frac{1}{3141}) − 64 arctan (\frac{1}{12943})

{5,13,17,61,101}

\frac{π}{4} = 27 arctan (\frac{1}{239}) + 200 arctan (\frac{1}{307}) + 68 arctan (\frac{1}{4193}) +  (55)

+56 arctan (\frac{1}{12238}) − 68 arctan (\frac{1}{39307}) + 112 arctan (\frac{1}{58911})

{5,13,29,97,421}

\frac{π}{4} = 95 arctan (\frac{1}{239}) + 44 arctan (\frac{1}{515}) + 32 arctan (\frac{1}{682}) +  (56)

+176 arctan (\frac{1}{782}) + 88 arctan (\frac{1}{4030}) + 112 arctan (\frac{1}{12943})

{5,13,61,101,401}

\frac{π}{4} = 183 arctan (\frac{1}{255}) + 44 arctan (\frac{1}{515}) − 56 arctan (\frac{1}{682}) +  (57)

+88 arctan (\frac{1}{1303}) + 7 arctan (\frac{1}{2943}) − 71 arctan (\frac{1}{12943})

{5,13,41,61,101}

\frac{π}{4} = 127 arctan (\frac{1}{265}) + 100 arctan (\frac{1}{515}) + 88 arctan (\frac{1}{697}) −  (58)

−32 arctan (\frac{1}{1068}) + 39 arctan (\frac{1}{2436}) − 56 arctan (\frac{1}{173932})

{5,13,37,73,101}

\frac{π}{4} = 68 arctan (\frac{1}{268}) + 115 arctan (\frac{1}{307}) + 32 arctan (\frac{1}{447}) +  (59)

+95 arctan (\frac{1}{1143}) + 51 arctan (\frac{1}{34208}) + 39 arctan (\frac{1}{44179})

{5,13,17,29,53}

\frac{π}{4} = 183 arctan (\frac{1}{268}) + 32 arctan (\frac{1}{682}) + 95 arctan (\frac{1}{1568}) +  (60)

+44 arctan (\frac{1}{4662}) − 166 arctan (\frac{1}{12943}) − 51 arctan (\frac{1}{32807})

{5,13,17,61,89}
\frac{π}{4} = 337 arctan (\frac{1}{307}) − 193 arctan (\frac{1}{463}) + 151 arctan (\frac{1}{4193}) +  (61)

+305 arctan (\frac{1}{4246}) − 122 arctan (\frac{1}{39307}) − 83 arctan (\frac{1}{390112})

{5,13,17,29,97}
\frac{π}{4} = 100 arctan (\frac{1}{319}) + 127 arctan (\frac{1}{378}) + 71 arctan (\frac{1}{557}) −  (62)

−15 arctan (\frac{1}{1068}) + 66 arctan (\frac{1}{2943}) + 44 arctan (\frac{1}{478707})

{5,13,17,41,73}
\frac{π}{4} = 322 arctan (\frac{1}{577}) + 76 arctan (\frac{1}{682}) + 139 arctan (\frac{1}{1393}) +  (63)

+156 arctan (\frac{1}{12943}) + 132 arctan (\frac{1}{32807}) + 44 arctan (\frac{1}{1049433})

{5,13,61,89,197}

62 \frac{π}{4} = 537 arctan (\frac{1}{11}) + 43 arctan (\frac{1}{5357}) − 52 arctan (\frac{1}{5507}) +  (64)

+6 arctan (\frac{1}{12943}) + 138 arctan (\frac{1}{17923}) + 112 arctan (\frac{1}{32807})

+26 arctan (\frac{1}{157318})

{5,13,37,61,89,97}

\frac{π}{4} = 44 arctan (\frac{1}{515}) + 95 arctan (\frac{1}{538}) + 127 arctan (\frac{1}{682}) +  (65)

+176 arctan (\frac{1}{782}) + 95 arctan (\frac{1}{1068}) + 88 arctan (\frac{1}{4030}) +

+17 arctan (\frac{1}{12943})

{5,13,61,73,101,401}

\frac{π}{4} = 366 arctan (\frac{1}{577}) − 12 arctan (\frac{1}{682}) + 44 arctan (\frac{1}{1023}) +  (66)

+183 arctan (\frac{1}{1393}) − 44 arctan (\frac{1}{5832}) + 24 arctan (\frac{1}{12943}) −

−88 arctan (\frac{1}{6826318})

{5,13,61,229,197,457}

\frac{π}{4} = 215 arctan (\frac{1}{682}) + 827 arctan (\frac{1}{1568}) − 139 arctan (\frac{1}{4662}) −  (67)

−349 arctan (\frac{1}{12943}) + 132 arctan (\frac{1}{32807}) − 366 arctan (\frac{1}{83270}) −

−183 arctan (\frac{1}{1493208})

{5,13,17,61,89,233}

\frac{π}{4} = 537 arctan (\frac{1}{746}) + 44 arctan (\frac{1}{5357}) + 259 arctan (\frac{1}{5507}) +  (68)

+156 arctan (\frac{1}{12943}) − 171 arctan (\frac{1}{17923}) + 227 arctan (\frac{1}{32807})

+139 arctan (\frac{1}{157318})

{5,13,37,61,89,97}

\frac{π}{4} = 1074 arctan (\frac{1}{1568}) + 657 arctan (\frac{1}{4662}) + 183 arctan (\frac{1}{5357}) −  (69)

−779 arctan (\frac{1}{12943}) − 32 arctan (\frac{1}{17923}) − 449 arctan (\frac{1}{32807}) +

+398 arctan (\frac{1}{390112})

{5,13,17,61,89,97}

\frac{π}{4} = 1587 arctan (\frac{1}{2852}) + 295 arctan (\frac{1}{4193}) + 593 arctan (\frac{1}{4246}) +  (70)

+359 arctan (\frac{1}{39307}) + 481 arctan (\frac{1}{55603}) + 625 arctan (\frac{1}{211050}) −

−708 arctan (\frac{1}{390112})

{5,13,17,29,97,433}

primes := subset of 5,13,17,29,37,41,53,61,73,89,97,101,109,113

\frac{π}{4} = 4 arctan (\frac{1}{5}) − arctan (\frac{1}{239})  (71)

{13}

\frac{π}{4} = 12 arctan (\frac{1}{18}) + 8 arctan (\frac{1}{57}) − 5 arctan (\frac{1}{239})  (72)

{5,13}

\frac{π}{4} = 44 arctan (\frac{1}{57}) + 7 arctan (\frac{1}{239}) − 12 arctan (\frac{1}{682}) + 24 arctan (\frac{1}{12943})  (73)

{5,13,61}

\frac{π}{4} = 88 arctan (\frac{1}{192}) + 39 arctan (\frac{1}{239}) + 100 arctan (\frac{1}{515})  (74)

−32 arctan (\frac{1}{1068}) − 56 arctan (\frac{1}{173932})

{5,13,73,101}

\frac{π}{4} = 100 arctan (\frac{1}{319}) + 127 arctan (\frac{1}{378}) + 71 arctan (\frac{1}{557})  (75)

−15 arctan (\frac{1}{1068}) + 66 arctan (\frac{1}{2943}) + 44 arctan (\frac{1}{478707})

{5,13,17,41,73} cf. formula 63

\frac{π}{4} = 1074 arctan (\frac{1}{1568}) + 657 arctan (\frac{1}{4662}) + 183 arctan (\frac{1}{5357})  (76)

−779 arctan (\frac{1}{12943}) − 32 arctan (\frac{1}{17923}) − 449 arctan (\frac{1}{32807})

+398 arctan (\frac{1}{390112})

{5,13,17,61,89,97} cf. formula 70

\frac{π}{4} = 1074 arctan (\frac{1}{4246}) + 1257 arctan (\frac{1}{5357}) + 1731 arctan (\frac{1}{6107})  (77)

+295 arctan (\frac{1}{12943}) + 625 arctan (\frac{1}{19703}) − 481 arctan (\frac{1}{32807})

−1042 arctan (\frac{1}{39307}) + 398 arctan (\frac{1}{390112})

{primes}

\frac{π}{4} = 7162 arctan (\frac{1}{12943}) + 3796 arctan (\frac{1}{32807}) + 2558 arctan (\frac{1}{34208})  (78)

+2729 arctan (\frac{1}{44179}) − 708 arctan (\frac{1}{51387}) + 2192 arctan (\frac{1}{114669})

−2805 arctan (\frac{1}{157318}) − 3696 arctan (\frac{1}{485298}) − 2407 arctan (\frac{1}{24208144})

{primes}

\frac{π}{4} = 2805 arctan (\frac{1}{5257}) − 398 arctan (\frac{1}{9466}) + 1950 arctan (\frac{1}{12943})  (79)

+1850 arctan (\frac{1}{34208}) + 2021 arctan (\frac{1}{44179}) + 2097 arctan (\frac{1}{85353}) +1484 arctan (\frac{1}{114669}) + 1389 arctan (\frac{1}{330182}) + 808 arctan (\frac{1}{485298})

{5,13,17,29,37,41,53,61} (Gauss)

\frac{π}{4} = 50539 arctan (\frac{1}{51387}) + 1555 arctan (\frac{1}{114669})  (80)

−6601 arctan (\frac{1}{157318}) − 20678 arctan (\frac{1}{390112}) − 5617 arctan (\frac{1}{485298})

−64126 arctan (\frac{1}{617427}) + 10958 arctan (\frac{1}{1984933}) − 30569 arctan (\frac{1}{3449051})

+23407 arctan (\frac{1}{22709274}) + 25433 arctan (\frac{1}{24208144})

{primes}

\frac{π}{4} = 36462 arctan (\frac{1}{390112}) + 135908 arctan (\frac{1}{485298})  (81)

+274509 arctan (\frac{1}{683982}) − 39581 arctan (\frac{1}{1984933})

+178477 arctan (\frac{1}{2478328}) − 114569 arctan (\frac{1}{3449051})

−146571 arctan (\frac{1}{18975991}) + 61914 arctan (\frac{1}{22709274})

−69044 arctan (\frac{1}{24208144}) − 89431 arctan (\frac{1}{201229582})

−43938 arctan (\frac{1}{2189376182})

{5,13,17,29,37,53,61,89,97,101}

\frac{π}{4} = 446879 arctan (\frac{1}{683982}) + 172370 arctan (\frac{1}{1635786})  (82)

−193720 arctan (\frac{1}{1984933}) + 369078 arctan (\frac{1}{2478328})

+18231 arctan (\frac{1}{3014557}) + 21339 arctan (\frac{1}{3449051})

−154139 arctan (\frac{1}{6225244}) − 110109 arctan (\frac{1}{18975991})

+80145 arctan (\frac{1}{22709274}) − 223183 arctan (\frac{1}{24208144})

−107662 arctan (\frac{1}{201229582}) − 216308 arctan (\frac{1}{2189376182})

{primes}

\frac{π}{4} = 872408 arctan (\frac{1}{1984933}) + 619249 arctan (\frac{1}{2298668})  (83)

+369078 arctan (\frac{1}{2478328}) + 18231 arctan (\frac{1}{3014557})

−1217159 arctan (\frac{1}{5033696}) + 911989 arctan (\frac{1}{6225244})

+783649 arctan (\frac{1}{18975991}) − 70886 arctan (\frac{1}{22709274})

−374214 arctan (\frac{1}{24208144}) − 1044789 arctan (\frac{1}{168623905})

+339217 arctan (\frac{1}{201229582}) − 446879 arctan (\frac{1}{284862638})

+402941 arctan (\frac{1}{2189376182})

{primes}