366 HlSTOKY OF THE THEOEY OF NtJMBEBS. [€HAP. XIV Ge*rardin76 gave a note on his machine to factor large numbers, especially those of the form 2x4— 1. FACTORING BY METHOD OF FINAL DIGITS. Johann Tessanek80 gave a tedious method of factoring N, not divisible by 2, 3, or 5, when JV/10 is within the limits of a factor table. For example, let 2Vr=10a+l; its factors end in 1,1 or 3,7 or 9,9. To treat one of the four cases, consider a factor lQx+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 extension81 to divisors 100:c+10/+0r. Again, to factor N=2a+l, given a table extending to N/2, note that if 2x+l is a divisor of Ny it divides a—x, which falls in the table. F. J. Studnicka82 quoted the last result. N. Beguelin83 would factor N=4p+3 by considering the final digit of IT = (j\T — ll)/4 and hence find the proper line in an auxiliary 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 JV. Georg Simon Klligel84 noted that a number, not divisible by 2, 3 or 5, is of the form 30z+m (m = l, 7, 11, 13, 17, 19, 23, 29). Suppose 10007 = (30x+m)(30y+n). Then (m, n) = (l, 17), (7, 11), (13, 29) or (19, 23). For w= 1, n= 17, we get 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=±l, he could factor any number within the limits of the table ! Sebastiano Canterzani85 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. Niegemann85a used the same method. Anton Niegemann86 gave a method of computing a table of squares arranged according to the last two digits. Thus, if A 76 = (lOx — 6)2, then 78Assoc. frang. avanc. sc., 43, 1914, 26-8. Proc. Fifth Internat. Congress, II, 1913, 572-3; Brit. Assoc. Reports, 1912-3, 405. 80Abhandl. einer Privatgesellschaft in Bohmen, zur Aufnahme der Math., Geschichte, . . . , Prag, I, 1775, 1-64. 81M. Cantor, Geschichte Math., 4, 1908, 179. ^Casopis, 14, 1885, 120 (Fortschr. der Math., 17, 1885, 125), "Nouv. M&n. Ac. Berlin, anne"e 1777 (1779), 265-310. wLeipziger Magazinfur reineu. angewandte Math, (eds., J.Bernoulli und Hindenburg), 1, 1787, 199-216. "Memorie dell' Istituto Nazionale Italiano, Classe di Fis. e Mat., Bologna, 2, 1810, II, 445-476. ^Entwickelung . . . Theilbarkeit, Jahresber. Kath. Gymn. Koln., 1847-8, 23. "Archiv Math. Phys., 45, 1866, 203-216 .