Chap. XIV] 
Methods of Factoring. 
369 
in view of the theorem of Lagrange: If x^—Ry'^=^D has relatively prime 
integral solutions x, y, where D< \/r, then D is a denominator of a com- 
plete quotient in the expansion of \/r as a continued fraction. 
Factoring by use of Various Moduli. 
C. F. Gauss^^° gave a "method of exclusion," based on the use of various 
small moduli, to express a given number in a given form mx^+ny^. 
V. Bouniakowsky^^^ noted that information as to the prime factors of a 
number N may be obtained by comparing the solution x = (l){N) of 2''=1 
(mod N) with the least positive solution x = a found by a direct process such 
as the following : Since 2" = NK+l, multiply the given N by the unknown 
K, each expressed in the binary scale (base 2), add 1 and equate the result to 
10. . .0. The digits of K are found seriatim and very simply. 
H. J. WoodalP^^ expressed the number N to be factored in the form 
^a_|_^6_^ . . . +r, where r< 1000, while a, /3, . . . are small, but not necessarily 
distinct. Hence the residues of N with respect to various moduli are 
readily found by tables of residues. 
F. Landry^^^ employed the method of exclusion by small moduli. 
D. Biddle^^^ investigated factors 2Ap + l by using moduli A^, 4A^. 
C. E. Bickmore, A. Cunningham and J. CuUen^^^ each treated the large 
factor of 10^^+1 by use of various moduli, and proved it is prime. 
J. Cullen"^" gave an effective graphical process to factor numbers by 
the use of various moduli; the numbers to be searched for in a diagram 
are all small. 
Alfred Johnsen^^^ used Rt(p) to denote the numerically least residue of 
p modulo t. Then, for every p, t, k, 
[Rmf+Ri(p-k')^Rt(p) (modO. 
If Hs a factor of the given number p, the left member will be divisible by t. 
In practice take k^ to be the nearest square to p, larger or smaller. For 
example, let p = 4699, k^ = 4624 = 68^ p-k'' = 75. Then 
t 
[^.(68)P 
Rt(75) 
Sum 
7 
13 
37 
4 
9 
36 
-2 
-3 
"i 
2 
6 
37 
Thus 37 is the least factor of p. 
""DJsq. Arith., 1801, Arts. 323-6. 
i"M6m. Ac. Sc. St. P^tersbourg, Math.-Phys., (6), 2, 1841, 447-69. Extract in Bull. Ac. Sc, 6, 
p. 97. Cf. Nordlund.i" 
"2Math. Quest. Educat. Times, 70, 1899, 68-71; 71, 1899, 124. 
"sproc^d^s nouveaux . . . , Paris, 1859. Cf. A. Aubry,"^ pp. 214-7. 
i"Mess. Math., 30, 1900-1, 98, 190. Math. Quest. Educat. Times, 74, 1901, 147-152. 
"'Math. Quest. Educat. Times, 72 1900, 99-103. 
"sa/bid., 73, 1900, 133-5; 75, 1901, 10^4. Proc. London Math. Soc, 34, 1901-2, 323-334; 
(2), 2, 1905, 138-141. "'Nyt Tidsskrift for Mat., 15 A, 1904, 109-110. 
