214 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VII v=0). Separate the residues modulo p of kq, for /b=l, 2,.. ., (p — into three sets: and form the differences rat- = p~ k From the set 1,. . ., (p — 1)/5, delete the Ti and m*; there remain v numbers v{. If yt is a root of s^s^ (mod p), then ofeg (mod p) is solvable if and only if ( — 1)^1. . .y^l (mod p), where 6 is the g. c. d. of n and p — 1. P. Seelhoff187 gave the known cases in which ofer (mod p) can be solved explicitly [Lagrange,153 Legendre156]. In the remaining cases, one uses Gauss' method of exclusion, the process of Desmarest,37 or, with SeelhofT, use various quadratic residues of p (ibid., p. 306). Here x2=41 (mod 120097) is treated. A. Berger188 considered a quadratic congruence reducible to #2= D (mod 4n), where D=0 or 1 (mod 4). If D is prime to n, the number of roots is where p ranges over the distinct prime factors of n, while d and di range over the pairs of complementary divisors of n, and f d — 0 or 1 according as d has a square factor or not. If g(nm)=g(n)g(m) for all integers n, m, and 0(1) = 1, where n ranges over all positive integers. Mean values are found : where A is a fundamental discriminant according to Kronecker, X, Xi are finite for all njs, and p ranges over all primes. G. Wertheim189 presented the theory of rr2=a (mod m). U. Marcolongo190 treated z2-f PE=O (mod p) in the usual manner when explicit solutions are known. Next, from a particular set of solutions x, y of 3?+pmy+P = Qf where p is a prime >2, we get the solution =*=£!=£-- pmy[ai.. .an_i] (mod pw+1) of Xiz+pm+ly1+P = Q, where [al.. .an_i] is the numerator of next to the last convergent to the continued fraction for pm/(2x). The method is Serret's, Alg. Supe"r., II. For p = 2 the results obtained are the same as in Dirichlet's Zahlentheorie, §36. 187Zeitschrift Math. Phya., 31, 1886, 378-80. 1886fversigt K. Vetenskaps-Ak. Forhandlingar, Stockholm, 44, 1887, 127-153. Nova Acta regiae soc. ac. Upsalensis, (3), 12, 1884. l«»Elemente der Zahlentheorie, 1887, 182-3, 207-217. 190Giornale di Mat., 25, 1887, 161-173.