370 History of the Theory of Numbers. [Chap, xiv 
K. P. Nordlund^^^ would use the exponent e to which 2 belongs modulo 
iV [Bouniakowsky^^^]. For A^ = 91, e = 12 is not a divisor of iV — 1, so that 
A^ is composite, and we expect the factor 13. 
L. E. Dickson"^ found the factors of 56'±1, 26'Hl, 34^^-}-l, 52^Hl 
by an expeditious method. For example, each factor of 
is =l(mod 14). Let 6 = (l + 14A;)(l + 14A'i). Then 
A'+;i-i + 14A'A-i=4Ar, /:+A'i = 4+14/1. 
Thus h and A-j are the roots of a quadratic whose discriminant Q is of the 
second degree in In. By use of various moduli which are powers of small 
primes, the form of h is limited step by step, until finally at most a half 
dozen values of h remain to be tested directly. 
L. E. Dickson^ ^® gave further illustrations of the last method. 
J. Schatunovsky^^° reduced to a minimum the number of trials in 
Gauss'^^° method of exclusion, taking the simplest case m = \. He gave 
theorems on the linear forms of the factors of a"~H-D6^, which lead easily to 
all its odd factors when D is an odd prime. 
H. C. Pocklington^^^ would use Fermat's theorem to tell whether A" is 
prime or composite. Choose an integer x and find the least positive residue 
of x^"^ modulo A''; if p^l, A'' is composite. But if it be unity, let p be a 
prime factor (preferably the largest) of A^— 1 and contained a times in it. 
Find the remainder r when x"* is divided by A', where 7n = (A' — l)/p. If 
Tt^I, let 6 be the g. c. d. of r— 1 and A^. If 6>1, we have a factor of A^. 
If 6 = 1, all prime factors of A^ are of the form Ap^ + l. But if r = 1, replace 
m by mlq^ where q is any prime factor of m and proceed as before. 
D. Biddle^^^" made use of various small moduli. 
A. Gerardin^^^'' used various moduli to factor 77073877. 
See papers 14, 15, 21, 22, 48, 65. 
Factoring Into Two Numbers 6n±l. 
G. W. Kraft^" ^^^^^ ^hat 6a+l = (6w+l)(6n+l) implies 
_ a—m 
^~6m + l' 
Find which 7?? = 1, 2, 3, . . . makes n an integer. 
Ed. BartP^ tested 6-186+5 for a prime factor less than 31, just less 
than its square root, by noting that 186, 185, 184, 183, 182 are not divisible 
by 5, 11, 17, 23, 29, respectively; while the last of 7, 13, 19 is a factor. 
'"Gotcborps Kunpcl. Vetenskaps. Hand!., (4), 1905, VII-VIII, pp. 21-4. 
»«Amer. Math. Monthly, 15, 1908, 217-222. "'Quar. Jour. Math., 40, 1909, 40-43. 
""Der Grosste GemeinschaftUche Teller von Algebr. Zahlen zweiter Ordnung, Diss. Strassburg, 
Leipzig, 1912. >"Proc. Cambridge Phil. Sec, 18, 1914-5, 29-30. 
"'"Math. Quest. Educat. Times, (2), 25, 1914, 43-6. 
"'"L'enseignement math., 17, 1915, 244-5. 
'"Novi Comm. Ac. Petrop., 3, ad annos 1750-1, 117-8. 
""Zur Theorie der Primzahlen, Progr. Mies, Pilsen, 1871. 
