Formulas of the Form
π
N
X
k =
mi arctan ( x1 )
i
4 i=1
by Jörg Arndt
[email protected]
the 1-term formula:
π
4
= arctan(1)
(1)
{}
2-term formulae:
π
4
= arctan ( 12 ) + arctan ( 13 )
(2)
{5} (Euler, 1706)
π
4
= 2 arctan ( 12 ) − arctan ( 71 )
(3)
{5} (Herman, 1706)
π
4
= 2 arctan ( 13 ) + arctan ( 71 )
(4)
{5} (Hutton, 1776)
π
4
1 )
= 4 arctan ( 15 ) − arctan ( 239
{13} (Machin, 1706)
(5)
3-term formulas:
π
4
1)
= 2 arctan ( 12 ) − arctan ( 19 ) − arctan ( 32
(6)
{5,41}
π
2
1 ) + arctan ( 1 )
= 3 arctan ( 12 ) + 2 arctan ( 13
38
(7)
{5,17}
π
2
1 ) − arctan ( 1 )
= 5 arctan ( 13 ) − 2 arctan ( 68
117
(8)
{5,37}
π
4
1 ) + arctan ( 1 )
= 3 arctan ( 14 ) + arctan ( 20
1985
(9)
{17,401}
π
4
1 ) − 2 arctan ( 1 )
= 5 arctan ( 17 ) + 2 arctan ( 23
182
(10)
{5,53}
π
4
1 ) − 2 arctan ( 1 )
= 5 arctan ( 17 ) + 2 arctan ( 26
2057
(11)
{5,677}
π
4
1 ) + 2 arctan ( 1 )
= 5 arctan ( 17 ) + 2 arctan ( 27
1068
(12)
{5,73}
π
4
1 ) + 2 arctan ( 1 )
= 5 arctan ( 17 ) + 2 arctan ( 28
443
(13)
{5,157}
π
4
1 ) + 2 arctan ( 1 )
= 5 arctan ( 17 ) + 2 arctan ( 43
68
(14)
{5,37}
π
4
1 ) + 2 arctan ( 1 )
= 5 arctan ( 17 ) + 4 arctan ( 53
4443
(15)
{5,181}
π
4
1 ) + arctan ( 1 )
= 6 arctan ( 18 ) + 2 arctan ( 57
239
(16)
{5,13} (Størmer, 1896)
π
4
1 ) − arctan ( 1 ) − 4 arctan ( 1 )
= 8 arctan ( 10
239
515
(17)
{13,101} (Klingstierna)
π
4
1 ) + 8 arctan ( 1 ) − 5 arctan ( 1 )
= 12 arctan ( 18
57
239
{5,13} (Gauss)
(18)
4-term formulas:
π
4
1 ) + 2 arctan ( 1 ) + 2 arctan ( 1 )
= 3 arctan ( 17 ) + 4 arctan ( 13
55
123
(19)
{ 5,17,89}
2
π
4
1 ) + arctan ( 1 ) + 2 arctan ( 1 )
= 11 arctan ( 17 ) + 4 arctan ( 443
1393
11018
(20)
{5,157,197}
5
π
4
1 ) − 9 arctan ( 1 ) − 16 arctan ( 1 ) − 12 arctan ( 1 )
= 44 arctan ( 11
239
682
12943
(21)
{5,13,61}
π
4
1 ) − arctan ( 1 ) − 4 arctan ( 1 ) − 4 arctan ( 1 )
= 12 arctan ( 15
239
580
1710
(22)
{ 13,113,229}
π
4
1 ) + 2 arctan ( 1 ) + 7 arctan ( 1 ) − 4 arctan ( 1 )
= 10 arctan ( 17
38
41
157
(23)
{5,17,29}
π
4
1 ) + 8 arctan ( 1 ) − 5 arctan ( 1 ) − 12 arctan ( 1 )
= 12 arctan ( 17
57
239
307
(24)
{5,13,29}
π
4
1 ) + 8 arctan ( 1 ) + 3 arctan ( 1 ) − 8 arctan ( 1 )
= 12 arctan ( 18
73
239
2943
(25)
{5,13,41}
π
4
1 ) + 3 arctan ( 1 ) + 5 arctan ( 1 ) + 8 arctan ( 1 )
= 12 arctan ( 18
70
99
307
(26)
{5,13,29} (Bennet)
π
4
1 ) + 8 arctan ( 1 ) + 3 arctan ( 1 ) + 8 arctan ( 1 )
= 12 arctan ( 18
99
239
307
(27)
{5,13,29} (D.H. Lehmer, 1938)
π
4
1 ) + 8 arctan ( 1 ) + 11 arctan ( 1 ) − 8 arctan (
1
= 12 arctan ( 18
109
239
6826318 )
(28)
{5,13,457}
3
π
4
1 ) − 23 arctan ( 1 ) + 8 arctan ( 1 ) − 16 arctan ( 1 )
= 44 arctan ( 18
239
682
12943
(29)
{5,13,61}
π
4
1 ) + 3 arctan ( 1 ) + 4 arctan ( 1 ) + 4 arctan ( 1 )
= 16 arctan ( 21
239
421
1985
(30)
{13,17,401}
π
4
1 ) + 7 arctan ( 1 ) + 12 arctan ( 1 ) + 2 arctan ( 1 )
= 10 arctan ( 22
41
75
4193
(31)
{5,29,97} (Størmer, 1896)
π
4
1 ) + 8 arctan ( 1 ) + 10 arctan ( 1 ) + 5 arctan ( 1 )
= 17 arctan ( 23
182
5118
6072
(32)
{5,53,373}
π
4
1 ) + 5 arctan ( 1 ) + 3 arctan ( 1 ) − 10 arctan ( 1 )
= 17 arctan ( 26
38
2057
8827
(33)
{5,17,677}
π
4
1 ) + 20 arctan ( 1 ) − 5 arctan ( 1 ) + 12 arctan ( 1 )
= 12 arctan ( 28
57
239
443
(34)
{ 5,13,157}
π
4
1 ) + 2 arctan ( 1 ) − 5 arctan ( 1 ) − 10 arctan ( 1 )
= 22 arctan ( 28
443
1393
11018
(35)
{ 5,157,197} (Escott)
π
4
1 ) − 4 arctan ( 1 ) + 7 arctan ( 1 ) − 12 arctan ( 1 )
= 24 arctan ( 29
57
239
12238
(36)
{5,13,421}
π
4
1 ) + 20 arctan ( 1 ) + 7 arctan ( 1 ) + 24 arctan ( 1 )
= 12 arctan ( 38
57
239
268
(37)
{ 5,13,17} (Gauss)
π
4
1 ) + 32 arctan ( 1 ) − 5 arctan ( 1 ) + 12 arctan ( 1 )
= 12 arctan ( 49
57
239
110443
(38)
{5,13,1201}
π
4
1 ) + 20 arctan ( 1 ) − 5 arctan ( 1 ) + 12 arctan ( 1 )
= 24 arctan ( 53
57
239
4443
(39)
{5,13,281}
π
4
1 ) + 24 arctan ( 1 ) + 12 arctan ( 1 ) − 5 arctan ( 1 )
= 20 arctan ( 57
68
117
239
(40)
{5,13,37}
π
4
1 ) + 7 arctan ( 1 ) − 12 arctan ( 1 ) + 24 arctan ( 1 )
= 44 arctan ( 57
239
682
12943
{5,13,61} (Størmer, 1896)
(41)
5-term formulas:
π
2
1 ) + arctan ( 1 ) +
= 11 arctan ( 71 ) + 2 arctan ( 99
1393
(42)
1 ) − 2 arctan (
1
−2 arctan ( 182
1264557 )
{5,13,29,53,197}
π
4
1 ) + 12 arctan ( 1 ) + 2 arctan ( 1 ) −
= 22 arctan ( 29
392
1068
(43)
1 ) − 10 arctan ( 1 )
−5 arctan ( 1393
5502
{5,73,197,421}
π
4
1 ) + 17 arctan ( 1 ) + 7 arctan ( 1 ) −
= 22 arctan ( 32
182
818
(44)
1 ) − 12 arctan (
1
−5 arctan ( 1303
3014557 )
{5,41,53,101}
π
4
1 ) − 12 arctan ( 1 ) + 39 arctan ( 1 ) +
= 68 arctan ( 99
117
239
(45)
1 ) − 24 arctan ( 1 )
+20 arctan ( 307
882
{5,13,29,37}
π
4
1 ) + 27 arctan ( 1 ) + 32 arctan ( 1 ) +
= 56 arctan ( 99
239
307
(46)
1 ) − 12 arctan ( 1 )
+12 arctan ( 4193
39307
{5,13,29,97}
π
4
1 ) + 95 arctan ( 1 ) − 12 arctan ( 1 ) +
= 44 arctan ( 109
239
682
(47)
1 ) − 44 arctan (
1
+24 arctan ( 12943
6826318 )
{5,13,61,457}
π
4
1 ) + 7 arctan ( 1 ) − 44 arctan ( 1 ) +
= 88 arctan ( 111
239
515
(48)
1 ) + 24 arctan ( 1 )
+32 arctan ( 682
12943
{5,13,61,101}
π
4
1 ) + 51 arctan ( 1 ) + 32 arctan ( 1 ) +
= 88 arctan ( 172
239
682
(49)
1 ) + 68 arctan ( 1 )
+44 arctan ( 5357
12943
{5,13,61,97} (Størmer, 1896)
π
4
1 ) + 39 arctan ( 1 ) + 100 arctan ( 1 ) −
= 88 arctan ( 192
239
515
1 ) − 56 arctan ( 1 )
−32 arctan ( 1068
173932
{5,13,73,101}
(50)
6-term formulas:
134
π
4
1 ) + 12 arctan ( 1 ) +
= 227 arctan ( 12 ) − 13 arctan ( 682
2059
(51)
1 ) − 6 arctan ( 1 ) + 20 arctan ( 1 )
+24 arctan ( 2943
5357
12943
{5,13,41,61,97}
3
π
4
1 ) + 81 arctan ( 1 ) + 4 arctan ( 1 ) −
= 200 arctan ( 99
239
4193
(52)
1 ) − 4 arctan ( 1 ) − 64 arctan ( 1 )
−32 arctan ( 12238
39307
58911
{5,13,29,97,421}
π
4
1 ) + 71 arctan ( 1 ) + 24 arctan ( 1 ) +
= 100 arctan ( 216
265
1068
(53)
1 ) + 44 arctan ( 1 ) + 56 arctan ( 1 )
+39 arctan ( 2436
5507
6962
{5,13,37,73,97}
π
4
1 ) + 176 arctan ( 1 ) + 44 arctan ( 1 ) +
= 7 arctan ( 239
293
616
(54)
1 ) + 132 arctan ( 1 ) − 64 arctan ( 1 )
+32 arctan ( 682
3141
12943
{5,13,17,61,101}
π
4
1 ) + 200 arctan ( 1 ) + 68 arctan ( 1 ) +
= 27 arctan ( 239
307
4193
(55)
1 ) − 68 arctan ( 1 ) + 112 arctan ( 1 )
+56 arctan ( 12238
39307
58911
{5,13,29,97,421}
π
4
1 ) + 44 arctan ( 1 ) + 32 arctan ( 1 ) +
= 95 arctan ( 239
515
682
(56)
1 ) + 88 arctan ( 1 ) + 112 arctan ( 1 )
+176 arctan ( 782
4030
12943
{5,13,61,101,401}
π
4
1 ) + 44 arctan ( 1 ) − 56 arctan ( 1 ) +
= 183 arctan ( 255
515
682
(57)
1 ) + 7 arctan ( 1 ) − 71 arctan ( 1 )
+88 arctan ( 1303
2943
12943
{5,13,41,61,101}
π
4
1 ) + 100 arctan ( 1 ) + 88 arctan ( 1 ) −
= 127 arctan ( 265
515
697
(58)
1 ) + 39 arctan ( 1 ) − 56 arctan ( 1 )
−32 arctan ( 1068
2436
173932
{5,13,37,73,101}
π
4
1 ) + 115 arctan ( 1 ) + 32 arctan ( 1 ) +
= 68 arctan ( 268
307
447
(59)
1 ) + 51 arctan ( 1 ) + 39 arctan ( 1 )
+95 arctan ( 1143
34208
44179
{5,13,17,29,53}
π
4
1 ) + 32 arctan ( 1 ) + 95 arctan ( 1 ) +
= 183 arctan ( 268
682
1568
(60)
1 ) − 166 arctan ( 1 ) − 51 arctan ( 1 )
+44 arctan ( 4662
12943
32807
{5,13,17,61,89}
π
4
1 ) − 193 arctan ( 1 ) + 151 arctan ( 1 ) +
= 337 arctan ( 307
463
4193
1 ) − 122 arctan ( 1 ) − 83 arctan ( 1 )
+305 arctan ( 4246
39307
390112
(61)
{5,13,17,29,97}
π
4
1 ) + 127 arctan ( 1 ) + 71 arctan ( 1 ) −
= 100 arctan ( 319
378
557
(62)
1 ) + 66 arctan ( 1 ) + 44 arctan ( 1 )
−15 arctan ( 1068
2943
478707
{5,13,17,41,73}
π
4
1 ) + 76 arctan ( 1 ) + 139 arctan ( 1 ) +
= 322 arctan ( 577
682
1393
1 ) + 132 arctan ( 1 ) + 44 arctan (
1
+156 arctan ( 12943
32807
1049433 )
{5,13,61,89,197}
(63)
7-term formulas:
62
π
4
1 ) + 43 arctan ( 1 ) − 52 arctan ( 1 ) +
= 537 arctan ( 11
5357
5507
(64)
1 ) + 138 arctan ( 1 ) + 112 arctan ( 1 )
+6 arctan ( 12943
17923
32807
1
+26 arctan ( 157318 )
{5,13,37,61,89,97}
π
4
1 ) + 95 arctan ( 1 ) + 127 arctan ( 1 ) +
= 44 arctan ( 515
538
682
(65)
1 ) + 95 arctan ( 1 ) + 88 arctan ( 1 ) +
+176 arctan ( 782
1068
4030
1 )
+17 arctan ( 12943
{5,13,61,73,101,401}
π
4
1 ) − 12 arctan ( 1 ) + 44 arctan ( 1 ) +
= 366 arctan ( 577
682
1023
(66)
1 ) − 44 arctan ( 1 ) + 24 arctan ( 1 ) −
+183 arctan ( 1393
5832
12943
1
−88 arctan ( 6826318
)
{5,13,61,229,197,457}
π
4
1 ) + 827 arctan ( 1 ) − 139 arctan ( 1 ) −
= 215 arctan ( 682
1568
4662
(67)
1 ) + 132 arctan ( 1 ) − 366 arctan ( 1 ) −
−349 arctan ( 12943
32807
83270
1
−183 arctan ( 1493208
)
{5,13,17,61,89,233}
π
4
1 ) + 44 arctan ( 1 ) + 259 arctan ( 1 ) +
= 537 arctan ( 746
5357
5507
(68)
1 ) − 171 arctan ( 1 ) + 227 arctan ( 1 )
+156 arctan ( 12943
17923
32807
1
+139 arctan ( 157318 )
{5,13,37,61,89,97}
π
4
1 ) + 657 arctan ( 1 ) + 183 arctan ( 1 ) −
= 1074 arctan ( 1568
4662
5357
(69)
1 ) − 32 arctan ( 1 ) − 449 arctan ( 1 ) +
−779 arctan ( 12943
17923
32807
1
+398 arctan ( 390112 )
{5,13,17,61,89,97}
π
4
1 ) + 295 arctan ( 1 ) + 593 arctan ( 1 ) +
= 1587 arctan ( 2852
4193
4246
1 ) + 481 arctan ( 1 ) + 625 arctan ( 1 ) −
+359 arctan ( 39307
55603
211050
1
−708 arctan ( 390112 )
{5,13,17,29,97,433}
(70)
primes := subset of 5,13,17,29,37,41,53,61,73,89,97,101,109,113
π
4
1 )
= 4 arctan ( 15 ) − arctan ( 239
(71)
{13}
π
4
1 ) + 8 arctan ( 1 ) − 5 arctan ( 1 )
= 12 arctan ( 18
57
239
(72)
{5,13}
π
4
1 ) + 7 arctan ( 1 ) − 12 arctan ( 1 ) + 24 arctan ( 1 )
= 44 arctan ( 57
239
682
12943
(73)
{5,13,61}
π
4
1 ) + 39 arctan ( 1 ) + 100 arctan ( 1 )
= 88 arctan ( 192
239
515
(74)
1 ) − 56 arctan ( 1 )
−32 arctan ( 1068
173932
{5,13,73,101}
π
4
1 ) + 127 arctan ( 1 ) + 71 arctan ( 1 )
= 100 arctan ( 319
378
557
(75)
1 ) + 66 arctan ( 1 ) + 44 arctan ( 1 )
−15 arctan ( 1068
2943
478707
{5,13,17,41,73}
π
4
cf. formula 63
1 ) + 657 arctan ( 1 ) + 183 arctan ( 1 )
= 1074 arctan ( 1568
4662
5357
(76)
1 ) − 32 arctan ( 1 ) − 449 arctan ( 1 )
−779 arctan ( 12943
17923
32807
1 )
+398 arctan ( 390112
{5,13,17,61,89,97}
π
4
cf. formula 70
1 ) + 1257 arctan ( 1 ) + 1731 arctan ( 1 )
= 1074 arctan ( 4246
5357
6107
(77)
1 ) + 625 arctan ( 1 ) − 481 arctan ( 1 )
+295 arctan ( 12943
19703
32807
1
1
−1042 arctan ( 39307 ) + 398 arctan ( 390112 )
{primes}
π
4
1 ) + 3796 arctan ( 1 ) + 2558 arctan ( 1 )
= 7162 arctan ( 12943
32807
34208
(78)
1 ) − 708 arctan ( 1 ) + 2192 arctan ( 1 )
+2729 arctan ( 44179
51387
114669
1 ) − 3696 arctan ( 1 ) − 2407 arctan (
1
−2805 arctan ( 157318
485298
24208144 )
{primes}
π
4
1 ) − 398 arctan ( 1 ) + 1950 arctan ( 1 )
= 2805 arctan ( 5257
9466
12943
(79)
1 ) + 2021 arctan ( 1 ) + 2097 arctan ( 1 )
+1850 arctan ( 34208
44179
85353
1 ) + 1389 arctan ( 1 ) + 808 arctan ( 1 )
+1484 arctan ( 114669
330182
485298
{5,13,17,29,37,41,53,61} (Gauss)
π
4
1 ) + 1555 arctan ( 1 )
= 50539 arctan ( 51387
114669
1 ) − 20678 arctan ( 1 ) − 5617 arctan ( 1 )
−6601 arctan ( 157318
390112
485298
1 ) + 10958 arctan (
1
1
−64126 arctan ( 617427
1984933 ) − 30569 arctan ( 3449051 )
1
1
+23407 arctan ( 22709274
) + 25433 arctan ( 24208144
)
{primes}
(80)
π
4
1 ) + 135908 arctan ( 1 )
= 36462 arctan ( 390112
485298
(81)
1 ) − 39581 arctan (
1
+274509 arctan ( 683982
1984933 )
1
1
+178477 arctan ( 2478328
) − 114569 arctan ( 3449051
)
1
1
−146571 arctan ( 18975991
) + 61914 arctan ( 22709274
)
1
1
−69044 arctan ( 24208144 ) − 89431 arctan ( 201229582 )
1
−43938 arctan ( 2189376182
)
{5,13,17,29,37,53,61,89,97,101}
π
4
1 ) + 172370 arctan (
1
= 446879 arctan ( 683982
1635786 )
(82)
1
1
−193720 arctan ( 1984933
) + 369078 arctan ( 2478328
)
1
1
+18231 arctan ( 3014557 ) + 21339 arctan ( 3449051 )
1
1
) − 110109 arctan ( 18975991
)
−154139 arctan ( 6225244
1
1
+80145 arctan ( 22709274
) − 223183 arctan ( 24208144
)
1
1
−107662 arctan ( 201229582 ) − 216308 arctan ( 2189376182 )
{primes}
π
4
1
1
= 872408 arctan ( 1984933
) + 619249 arctan ( 2298668
)
1
1
) + 18231 arctan ( 3014557
)
+369078 arctan ( 2478328
1
1
−1217159 arctan ( 5033696
) + 911989 arctan ( 6225244
)
1
1
+783649 arctan ( 18975991 ) − 70886 arctan ( 22709274 )
1
1
−374214 arctan ( 24208144
) − 1044789 arctan ( 168623905
)
1
1
) − 446879 arctan ( 284862638
)
+339217 arctan ( 201229582
1
+402941 arctan ( 2189376182
)
{primes}
(83)