374 History of the Theory of Numbers. [Chap, xiv 
S. Bisman^"** noted that N is composite if and only if there exist two 
integers A, B such that A +2 B and A-\-2BN divide 2{N-1) and (A''-1)A, 
respectively. But there is no convenient maximum for the smaller integer 
B. To find the factor 641 of 2'^^ + l there are 16 cases. 
A. G^rardin^^^ gave a report on methods of factoring. 
J. A. Gmeiner,^^° to factor a, prime to 6, determined b and e so that 
9a = 166+e, 0^€<16. Let cu^ be the largest square <6 and set b = oi^-\-p, 
cr = p— CO. Hence 9a = 16(co — a:)(w+a;H-l)+r(a:), where 
t(x) = 16(7+€+16x(x+1). 
Since t{x)=t{x — 1)+S2x, we may rapidly tabulate the values of r(x) for 
x = 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 co+x + l is the 
largest square <a. To modify the process, use 4a = 76 + €. 
A. Reymond^^^ 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 ony = x/2. Since 17 is not the abscissa of an integral 
point on y = x/n for l<n< 17, 17 is a prime. [Mobius^^" of Ch. XIII.] 
A. J. Kempner^^^ found, by use of a figure perspective to Reymond's^^^ 
how to test the primality of numbers by means of the straight edge. 
D. Biddle and A. Cunningham^^^ factor a product A^ of two primes by 
finding A^i<A^ and N2>N such that N2-N = N-Ni+2, while each of Ni 
and A''2 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. 
"'Assoc. fraiiQ. avanc. sc, 41, 1912, 54-7. 
"oMonatshefte Math. Phys., 24, 1913, 3-26. 
»"L'enseignement math., 18, 1916, 332-5. 
'"Amer. Math. Monthly, 24, 1917, 317-321. 
'"Math. Quest, and Solutions, 3, 1917, 21-23. 
I 
