366 History of the Theory of Numbers. [Chap, xiv 
G^rardin^® gave a note on his machine to factor large numbers, espe- 
cially those of the form 2x^ — 1. 
Factoring by Method of Final Digits. 
Johann Tessanek^° gave a tedious method of factoring N, not divisible 
by 2, 3, or 5, when A^/10 is within the limits of a factor table. For example, 
let N=10a+l; its factors end in 1,1 or 3,7 or 9,9. To treat one of the 
four cases, consider a factor lOx+3, the quotient being 102+7. Then z is 
the quotient of a — 2 — 7x by lOx+3. Give to x the values 1, 2, . . ., and test 
a — 9 for the factor 13, a — 16 for 23, etc., by the factor table. He gave a 
lengthy extension^^ to di\'isors 100j+10/+gr. Again, to factor iV = 2a + l, 
given a table extending to N/2, note that if 2xH-l is a divisor of N, it 
di\'ides a—x, which falls in the table. F. J. Studnicka^^ quoted the last 
result. 
N. Beguelin^ would factor iV = 4p+3 by considering the final digit of 
TT = (A'' — 1 1)/4 and hence find the proper line in an auxihary table (pp. 291-2) , 
each line containing four fractional expressions. Proceed with each until 
we reach a fraction whose numerator is zero. Then its denominator is a 
factor of A^. 
Georg Simon KliigeP noted that a number, not divisible by 2, 3 or 5, 
is of the form 30x+m (m = l, 7, 11, 13, 17, 19, 23, 29). Suppose 10007 = 
(30x+w)(30!/+n). Then {m, n) = (l, 17), (7, 11), (13, 29) or (19, 23). 
For m = 1, n = 17, we get 
333-?/ 
^=3o^Ti7' ^<^' y<^' 
But X is not integral for y = 0, 1, 2, 3. 
Johann Andreas von Segner {ibid., 217-225) took two pages to prove 
that any number not divisible by 2 or 3 is of the form 6n='= 1 and noted that, 
given a table of the least prime factor of each 6n=tl, he could factor any 
number within the limits of the table! 
Sebastiano Canterzani^^ would factor 10^ + 1, by noting the last digits 
1, 1 or 3, 7 or 9, 9 of its factors. If one factor ends in 7, there are 10 
possibilities for the digit preceding 7; if one ends in 1 or 9, there are five 
cases; hence 20 cases in all. A. Niegemann^^" used the same method. 
Anton Niegemann^® gave a method of computing a table of squares 
arranged according to the last two digits. Thus, if A76 = {l0x — Qy, then 
™\s30c. fran?. avanc. sc, 43, 1914, 26-8. Proc. Fifth Internat. Congress, II, 1913, 572-3; 
Brit. Assoc. Reports, 1912-3, 405. 
••Abhandl. einer Privatgesellschaft in Bohmen, zur Aufnahme der Math., Geschichte, . . . , Prag, I, 
1775, 1-64. 
"M. Cantor, Geschichte Math., 4, 1908, 179. 
"Casopis, 14, 1885, 120 (Fortschr. der Math., 17, 1885, 125). 
"Nouv. Mdm. Ac. Berhn, ann^e 1777 (1779), 265-310. 
•*Leipziger Magazin fiir reine u. angewandte Math, (eds., J. Bernoulli und Hindenburg), 1, 1787, 
199-216. 
"Memorie dell' Istituto Xazionale Italiano, Classe di Fis. e Mat., Bologna, 2, 1810, II, 445-476. 
""Entwickelung . . . Theilbarkeit, Jahresber. Kath. Gymn. Kobi., 1847-8, 23. 
"Archiv Math. Phys., 45, 1866, 203-216 . 
