Chap. XIV] METHODS OF FACTORING. 359 
Then 2an+n^-r = s^ and m = {a-\-ny-s^. The method is the more rapid 
the smaller the difference of the two factors. 
M. Neumann^^ proved that this process of adding terms leads finally 
to a square and hence to factors, one of which may be 1. 
F. W. Lawrence^^ denoted the sum of the two factors of n by 2a and the 
difference by 2b, whence n = a^ — 6^. Let q be the remainder obtained by 
dividing n by a chosen prime p, and write down the pairs of numbers < p 
such that the product of two of a pair is congruent to q modulo p. If 
p = 7, q = S, the pairs are 1 and 3, 2 and 5, 4 and 6, whence 2a=4, or 3 
(mod 7). Using various primes p and their powers, we get limitations on 
a which together determine a. The work may be done with stencils. The 
method was used by Lawrence^^ to show that five large numbers are primes, 
including 10, 11 and 12 place factors of 3^^ — 1, 10^^ — 1, 10^^ — 1, respectively. 
The same examples were treated by other methods by D. Biddle.^^ 
A. Cunningham^'^ remarked that in computing by Busk's method a 
k for which {s+ky—N is a, square, we may use the method of Lawrence, 
just described, to limit greatly the number of possible forms of k. 
F. J. Vaes^^ expressed N in the form a^ — b"^ by use of the square a^ just 
>A^ and then increasing a by 1, 2, ... , and gave (pp. 501-8) an abbreviation 
of the method. He strongly recommended the method of remainders 
(p . 425) : If p is a factor oiG = h^ — g^, and iig={G — l)/2 has the remainder 
r when divided by p, then h={G+l)/2 must have the remainder r+1, 
so that p is a factor of 2r-\-l=G. For example, let G = 80047, whence 
^ = 200H23 = 20M99+24 = 202-198+27,.... 
For r = 24, 27, 32, . . . we see that 2r+l is not a multiple of 201, 202, . . . 
until we reach gr = 209-191 +p, p=104, 2p+ 1 = 209. Thus 209 divides G. 
P. F. Teilhet^^ wrote N = a^-b in the form (a+kY-P, where P = k^ 
-{-2ak-{-b. Give to k successive values 1, 2, . . . (by additions to P), until 
P becomes a square v^. To abbreviate consider the residues of P for small 
prime moduli. 
E. Lebon^° proceeded as had Teilhet^^ and then set /=a+A; — y. Then 
2kf={a-fy-b, 
and we examine primes /< a to see if k is an integer. 
M. Kraitchik^^ would express a given odd number A in the form 
if—x^ by use of various moduli p. Let A = r (mod p) and let Oi, . . . , a„ be the 
"Zeitschrift Math. Naturw. Unfcerricht, 27, 1896, 493-5; 28, 1897, 248-251. 
"Quar. Jour. Math., 28, 1896, 285-311. French transl., Sphinx-Oedipe, 5, 1910, 98-121, with 
an addition by Lawrence on g^ +1. 
isProc. Lond. Math. Soc, 28, 1897, 465-475. French transl., Sphinx-Oedipe, 5, 1910, 130-6. 
i^Math. Quest. Educat. Times, 71, 1899, 113-4; cf. 93-99. 
"/bid., 69, 1898, 111. 
i«Proc. Sect. Sciences Akad. Wetenschappen Amsterdam, 4, 1902, 326-336, 425-436, 501-8 
(EngUsh); Verslagen Ak. Wet., 10, 1901-2, 374-384, 474-486, 623-631 (Dutch). 
i^L'intermediaire des math., 12, 1905, 201-2. Cf. Sphinx-Oedipe, 1906-7, 49-50, 55. 
"Assoc, franc, av. sc, 40, 1911, 8-9. 
^^Sphinx-Oedipe, Nancy, Mai, 1911, num^ro special, pp. 10-16. 
