372 HlSTOBY OF THE THBOKY OF NUMBERS. [CHAP. XIV on dividing f by 2), 1 +2, 1 +2+2 under 1, 3, 5 of the first row for r = 1 ; in the lower TOW, insert 4 (the quotient), 4—1, 4— 2. To f actor /2— c, locate the column headed by the given c; thus, for c = 3, the factors are s = 6, r = 1 and s = 3, r = 2. Since c = 2 occurs only in the first row, 9 — 2 is prime. Joubin,133 J. P. Kulik,134 0. V. Kielsen,136 and G. K Winter136 published papers not accessible to the author. E. Lucas gave methods of factoring and tests for primes (Ch. XVII). D. Biddle137 wrote the proposed number N in the form $2+.A, where S2 is the largest square < N. Write three rows of numbers, the first beginning with Ay or A —S if A >$; the second beginning with S (or £+1) and increasing by 1 ; the third beginning with S and decreasing by 1. Let An, Bn, Cn be the nth elements in the respective rows. Then except that, when An>Cn, we subtract Cn from An as often (say k times) as will leave a positive remainder, and then Bn — Bn^i+l+k. When we reach a value of n for which An=0, we have N-BnCn. For example, if 2V==589 = 242+13, the rows are 13 14 17 1 9 0 24 25 26 28 29 31 (factors 31, 19). 24 23 22 21 20 19 It may prove best to start with 2N instead of with -2V. O. Meissner138 reviewed many methods of factoring. R. W. D. Christie139 gave an obscure method by use of "roots." Christie140 noted that, if N=AB, A = (4bN+d2-d)/(2b), B = (4bN+d2+d)/2, d=a-bc, whence d2 = (B-6A)2. D. Biddle141 gave a method of finding the factors of N given those of N+l. Set L~N-l. Try to choose K and M so that KM ^ N+l and so that 1 +K is a factor of N. Since 2N = (1 +K) M +L - M, we will have L-M=(l+jK)m, whence 2N~(l+K)(M+m). For N=1829, N+l = 2-3-5-61. Take # = 30, M = 61. Then m = 57, M+w = 2-59, # = 31-59. He gave (ibid., p. 43) the theoretical test that N = S2+A is composite if the sum of r terms of A, N N is an integer for some value of r. 138Sur lea facteurs num&iques, Havre, 1831. »«Abh. K. Bohm. Geseli Wiss., 1, 1841 (2, 1842-3, 47, graphical determination of primes). 136Om et heel tals upplosning i factorer, Kjobenhavn, 1841. 136Madras Jour, Lit. Sc., 1886-7, 13. 137Me88. Math., 28, 1898-9, 116-20; Math. Quest. Educat. Times, 70, 1899, 100, 122; 75, 1901, 48; extension, (2), 29, 1916, 43-6. 138Math. Naturw. Blatter, 3, 1906, 97, 117, 137. 139Math. Quest. Educat. Times, (2), 12, 1907, 90-1, 107-8. ™Ibid., (2), 13, 1908, 42-3, 62-3. ™Ibid., (2), 14, 1908, 34. The process is well adapted to factoring 2^-1, (2), 23, 1913, 27-8.