214 History of the Theory of Numbers. [Chap, vii 
^ = 0). Separate the residues modulo p of kq, for k = l, 2, . . ., {p — l)/d, 
into three sets: .': 
P 5-1 i 
0<ri,. . ., r,<-<Si,. . ., s,<-^p<tu. • ., t^<p j 
and form the differences mi = p — t,. From the set 1,. . ., (p — 1)/5, delete 
the r, and ?n,; there remain v numbers i\. If ?/, is a root of s,?/,= yi (mod p), 
then x"=5 (mod p) is solvable if and only if ( — l)"?/!. . .2/„=l (mod p), 
where 5 is the g. c. d. of n and p — 1 . 
P. Seelhoff^^^ gave the known cases in which x^=r (mod p) can be 
solved explicitly [Lagrange, ^^^ Legendre^^^]. In the remaining cases, one 
uses Gauss' method of exclusion, the process of Desmarest,^^ or, with 
Seelhoff, use various quadratic residues of p {ibid., p. 306). Here x^=41 
(mod 120097) is treated. 
A. Berger^^^ considered a quadratic congruence reducible to a:^=D (mod 
4n), where D=0 or 1 (mod 4). If D is prime to n, the number of roots is 
^(D, in) = 2n{l + (f ) } = 2S (f ) f . = 22 (f ) f,, 
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 ^ = or 1 according as 
d has a square factor or not. If g{nm)=g{n)g{m) for all integers n, m, 
ands'(l) = l, 
zgy (Z), 4n)^(n) = 22 Q ^(n) -S Q ^(n) -^2 Q ^(n)^ 
where n ranges over all positive integers. Mean values are found : 
J,(?)*»«-;b5ti7S.?,©I«-" 
it=i TT h=i\n/n 
where A is a fundamental discriminant according to Kronecker, X, Xi are 
finite for all n's, and p ranges over all primes. 
G. Wertheim^^^ presented the theory of a;^=a (mod m). 
R. Marcolongo^®° treated x^-\-P=0 (mod p) in the usual manner when 
explicit solutions are known. Next, from a particular set of solutions 
X, y of x^+p'"?/+P = 0, where p is a prime >2, we get the solution 
=i=x,=x-p'"y[ai.. .o„_i] (mod p"'+') 
of Xi^-\-p"''^^yi-{-P = 0, where [ai. . .a„_i] is the numerator of next to the 
last convergent to the continued fraction for p"'/{2x). The method is 
Serret's, Alg. Sup^r., II. For p = 2 the results obtained are the same as in 
Dirichlet's Zahlentheorie, §36. 
i"Zeitschrift Math. Phys., 31, 1886, 378-80. 
"SQfversigt K. Vetenskaps-Ak. Forhandlingar, Stockholm, 44, 1887, 127-153. Nova Acta 
regise soc. sc. Upsalensis, (3), 12, 1884. 
"»Elemente der Zahlentheorie, 1887, 182-3, 207-217. ""Giomale di Mat., 25, 1887, 161-173. 
