374 HlSTOET OF THE THEORY OP NUMBEBS. [CHAP. XIV S. Bisman148 noted that N is composite if and only if there exist two integers A, B such that A+2B and A+2BN divide 2(JV-1) and (N-1)A, respectively. But there is no convenient maximum for the smaller integer B. To find the factor 641 of 232+l there are 16 cases. A. Ge*rardin149 gave a report on methods of factoring. J. A. Gmeiner,150 to factor a, prime to 6, determined b and € so that 9a = 166+€, 0^e<16. Let co2 be the largest square <b and set & = o>2+p} <r=p— co. Hence 9a=16(co — X)(W+X+I)+T(X), where Since T(aO=r(z — l)-f32z, we may rapidly tabulate the values of r(x) for £=0, 1, 2, . . .. If we reach the value zero, we have two factors of a. To prove that a is a prime, we need extend the table until oj+x-f 1 is the largest square <a. To modify the process, use 4a = 76+e. A. Reymond151 used the graphs of y=x/n (n=l, 2, 3, 5, . . .), marking on each the points with integral coordinates. He omitted y=x/4 since its integral points are on y—x/2. Since 17 is not the abscissa of an integral point on y^x/n for l<n< 17, 17 is a prime. [Mobius530 of Ch. XIII J A. J. Kempner162 found, by use of a figure perspective to Reymond's151, how to test the primality of numbers by means of the straight edge. D. Biddle and A. Cunningham163 factor a product N of two primes by finding Ni<N and N2>N such that N2-N=N-Nl+2, while each of NI and N2 is a. product of two even factors, the two smaller factors differing by 2 and the two larger factors differing by 2. "'Mathesis, (4), 2, 1912, 58-60. 149Assoc. frang. avanc. sc., 41, 1912, 54-7. lfi<)Monatsliefte Math. Phys., 24, 1913, 3-26. ^L'enseignement math., 18, 1916, 332-5. "2Amer. Math. Monthly, 24, 1917, 317-321. 153Math. Quest, and Solutions, 3, 1917, 21-23.