368 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xiv solutions if a is not a square) and testing x2 — 1 for a factor in common with a. Further, if ay+l=x2 does not hold for Ka?<a— 1, then a is a power of a prime and conversely [false if a= 10]. Marcker101 noted that if there are 2n terms in the period of andQ-0, Q'==a, Q'' P-l f—*3 then the nth P or its half is a factor of A. HA is a prime, then the nth Pis 2. J. G. Birch102 derived a factor of N from a solution x of x2 =Ny+ 1. The continued fraction for x/(N—x) is of the form 1 ! ! and AT is the continuant defined as the determinant with a0, &i, . . . , an_i, an, an_i,..., cti, a0 in the main diagonal, elements +1 just above this diagonal, elements —1 just below, and zeros elsewhere. Then the continuant with the diagonal a0, . . . , <V-i is a factor of N. W. W. R. Ball103 applied this method to a number of Mersenne.1 A. Cunningham104 noted that a set of solutions of y2—Dx2 — — 1 gives at sight factors of y2+l. M. V. Thielmann106 illustrated his method by factoring fc = 36343817. The partial denominators in the continued fraction for \/k are 1, 1, 2, 1, 1, 12056. Drop the last term and pass to the ordinary fraction 7/12. Hence set (12x+7)2=122?/-|-l. The least solution is x = 4, y = 2l. Using thepart of the period preceding the middle term 10 = 2, we get Hence f— 21w2 = 1 has the solution £ = 55. For a suitably chosen n, where q is the largest integer fg Q/2. Here n = 502 and the factors of k are 2-32rz,-h7and2-22n+3. D. N. Lehmer106 noted that if R = pq is a product of two odd factors whose difference is <2\/S, so that l(p—q)2<\*/rR, then *2-%2=K?-<z)2 has the integral solutions x= (p+q)/2, y — 1. Hence J(P"~<?)2 ig a denominator of a complete quotient in the expansion of \SR as a continued fraction, 101 Jour, fur Math., 20, 1840, 355-9. Cf. I'intermSdiaire des math., 20, 1913, 27-8. 102Mess. Math., 22, 1892-3, 52-5. 103/6td., p. 82-3. French transl., with Birch103, Sphinx-Oedipe, 1913, 86-9. ™Ibid., 35, 1905-6, 166-185; abst. in Proc. London Math. Soc., 3, 1905, xxii. 106Math. Annalen, 62, 1906, 401. 106Bull. Amer. Math. Soc., 13, 1906-7, 501-2. French transl., Sphinx-Oedipe, 6, 1911, 138-9.