368 History of the Theory of Numbers. [Chap, xiv 
solutions if a is not a square) and testing x" — 1 for a factor in common with a. 
Further, U ay-\-l=x'^ does not hold for l<x<a — 1, then a is a power of a 
prime and conversely [false if a = 10]. 
Marcker^*'^ noted that if there are 2n terms in the period of 
and Q = 0, Q' = a, Q" = a'P' -Q',. . .,^ 
p_i p/_ ^~Q'' ptf _ ^~Q"^ 
r — 1, r — p ' pf ) • ■ • i 
then the nth P or its half is a factor of A. If A is a prime, then the nth 
Pis 2. 
J. G. Birch^°- derived a factor of N from a solution x of x'^ = Ny+l. The 
continued fraction for x/{N —x) is of the form 
1 J_ J_ ill 
Go- 1 +01+02+ ■ • • +a2+ai+ao' 
and N is the continuant defined as the determinant with Oq, Oi,. . ., a„_i, 
On) On-i>- •■> Oi, Oo in the main diagonal, elements +1 just above this 
diagonal, elements —1 just below, and zeros elsewhere. Then the continu- 
ant with the diagonal ao> • • •> o„_i is a factor of N. 
W. W. R. BalP°^ appHed this method to a number of Mersenne.^ 
A. Cunningham^*^ noted that a set of solutions of if—Dx^ = — 1 gives at 
sight factors of 7/" + l. 
M. V. Thielmann^o^ illustrated his method by factoring /c = 36343817. 
The partial denominators in the continued fraction for \/^ are 1, 1, 2, 1, 1, 
12056. Drop the last term and pass to the ordinary fraction 7/12. Hence 
set (12x+7)^ = 12^i/+l. The least solution is x = 4, y = 2\. Using the part 
of the period preceding the middle term it» = 2, we get 
Y^ = -^, P = l, M = 2, Q = iyM+2P = 6, u = MQ = \2. 
Hence ^" — 21m^ = 1 has the solution < = 55. For a suitably chosen n, 
ifc = wV+2^n+21= (2g2n+^^ Uu^n+^-^X 
where q is the largest integer ^ Q/2. Here n = 502 and the factors of k are 
2-3"n + 7 and 2-22n+3. 
D. N. Lehmer^''^ noted that if jR = pg' is a product of two odd factors 
whose difference is <2y/R, so that Kp— 9)^<V7^, then 
x'-Rf = \{V-q? 
has the integral solutions x = (p+g')/2, y = \. Hence i(p — g')^ is a denomi- 
nator of a complete quotient in the expansion of y/R, as a continued fraction, 
""Jour, fur Math., 20, 1840, 355-9. Cf. I'interm6diaire dea math., 20, 1913, 27-8. 
J^Mcsa. Math., 22, 1892-3. 52-5. 
'o'/6id., p. 82-3. French transl., with Birch"^, Sphinx-Oedipe, 1913, 86-9. 
>"/6id., 35, 1905-6, 166-185; abat. in Proc. London Math. Soc. 3, 1905, xxii. 
'""Math. Annalen, 62, 1906, 401. 
Bull. Amer. Math. Soc, 13, 1906-7, 501-2. French transl., Sphinx-Oedipe, 6, 1911, 138-9. 
IM 
