54] BERNOULLIAN NUMBERS FROM #, 2 TO , 3 . 443 



The following extract from the calculations for B 31 supplies the further data 



which are required in making the similar calculations for B^ 2 . 

 Table of the products P? for n = 3l, and calculation of the quantities 7 31 and B 3l . 



3 



5091 



51 78275 



32629 66196 



119 34005 74495 



23247 08675 60794 



21 42517 90194 62108 



785 22331 21935 Il8l6 



8645 5I75I 16414 13997 



16774 64384 84868 767II 



26227 04351 88775 49492 30619 77419 37134 85683 Sum 



135 



I 71790 

 1385 12625 

 6 71133 42236 

 1813 75138 15056 



2 47319 34636 79761 

 147 61606 58640 58443 

 3102 83046 11851 35783 

 15767 53321 46412 01553 

 8498 13384 95367 52252 



27518 60498 94566 69639 20928 76040 64824 28921 Sum 

 197 93228 99666 11337=^31 



27518 60498 94566 69639 21126 69269 64490 40258 

 26227 04351 88775 49492 30619 77419 37134 85683 



63 ) 1291 56147 05791 20146 90506 91850 27355 54575 -31 

 20 50097 57234 78097 56992 17330 95672 31025 =/ 31 



Also = 7+l- 



Hence the numerator of B 31 is 123 00585 43408 68585 41953 03985 74033 86151 

 and the denominator is 6 



As a test, this numerator should be divisible by 31. 



By actual division we find the quotient to be 3 96793 07851 89309 20708 16257 60452 70521 

 without any remainder. Hence the test is satisfied. 



Table of the factors by which the quantities P r n for n = 3 1 must be multiplied 

 in order to find the corresponding quantities for n = 32. 



r= 7, Factor: 

 r=io, Factor: 

 r=i3, Factor: 

 j-=i6, Factor; 

 > - =i9, Factor: 

 r = 22, Factor^ 

 r=25, Factor 

 r=28, Factor 

 r = 3i, Factor 



65 . 64 _ 13 . 32 



51 . 50 

 .65.64 

 '45-44" 



65-64 

 '39- 38" 

 .65- 64. 

 ' 33 32 " 

 .65-64.. 



27 . 26 



.65-64. 

 "21 . 20" 

 .65-64 

 "15 . 14" 



.65-64. 

 - 9.8 - 



.65-64. 

 3 2 



5i 5 

 13 . 16 

 9- 

 5- 32 

 3- 19 



.65.2 

 33 



.5-32 



27 

 13 . 16 



21 



.I3-32 



'3-7 

 65. 8 



9 



. 65 - 32 

 3 



_65 . 64 _65 . 4 

 ~49- 48 ~ 49 3 



65 .64 65 . 32 



= 11, Factor = ^ 5-_2 2_ 



43 .42 43 . 21 



65 . 64 _ 65 . 16 



r=i4, Factor = 



r=i7, 



37 36 37-9 

 _6s . 64 _ 13. 32 

 31 . 30 31 . 3 



6? . 64 13.8 



= 20, Factor =~ "= 



25-24 5-3 



65 .64 65 . 32 



= 23, Factor =-^ -5==-* ^ 



19 . 18 19 . 9 



_65 . 64 _ 5 . 16 

 "13 . 12" 3 

 _6s . 64 _ 65. 32 

 7-6 7-3 



r=26, 



r=29 



r = a, 

 r=i2, 

 r=i5, 

 r=i8, 



_ , 65 . 

 Factor = - J 

 47- 



65 . 



Factor = 

 41 . 



65 . 



^ 



35 



29 . 



65 . 



Factor =- - 

 23 . 



r=27, 

 r=30, 



Factor =^- 

 II 



Factor s 6 ^- 



64 _ 65 . 32 

 46 47 23 

 64 _ 13 . 8 

 40" 41 

 64 __ 13 . 32 

 34^ 7- 17 

 64 _ 65 . 16 

 "28" 29. 7 

 64 _ 65 . 32 



22 23 . II 



64 _ 65 . 4 

 16~~I7~ 

 64 _ 13 . 32 

 10 ii 



64 

 ==13. 16 



562 



