Chap. VII] BiNOMIAL CONGEUENCES. 207 
case p = 8n+5; but in general no such direct solution is known, and it is 
best to represent some multiple of p by the form y'^+az^. 
If we have found 6 such that ^^+a is divisible by the prime p, not dividing 
a, we readily solve x^+a = (mod p"). For, from 
r^-\-as^ is divisible by p^. Now s is not divisible by p. Thus we may take 
r = sx+p''y, whence x^+a is divisible by p". [Cf. Tchebychef, Theorie der 
Congruenzen, §30.] 
The case of any composite modulus N is easily reduced to the preceding 
(end of Lagrange's^^^ paper). Legendre proved that, if N is odd and prime 
to a, the number of solutions of a:^+a = (mod N) is 2'"^ where i is the 
number of distinct prime factors of N; the same is true for modulus 2N. 
Henceforth let N be odd or the double of an odd number and let d be the 
g. c. d. of N and a. If d has no square factor, the congruence has 2'"^ roots, 
where i is the number of distinct odd prime factors of N not dividing a. 
But if d=o)\(/^, where co has no square factor, the congruence has 2'~V 
roots where i is the number of distinct odd prime factors of N/d. 
C. F. Gauss^" treated congruence (1) by the use of indices. However, 
we can give a direct solution (arts. 66-68) when a root is known to be con- 
gruent to a power of B. For, by (1) and x = B^, B^B^"". If therefore a 
relation of the last iy^e is known, a root of (1) is B''. The condition for 
the relation is l = A:n (mod t), where t is the exponent to which B belongs 
modulo p. It is shown that t must divide m = (p — l)/n. We may discard 
from m any factor of n; if the resulting number is m/q, the unique solution 
k of 1 = 1:11 (mod m/q) is the desired k. [Cf. Poinsot^^^] 
Gauss (arts. 101-5) gave the usual method of reducing the solution of 
x^= A (mod m) for any composite modulus to the case of a prime modulus 
and gave the number of roots modulo p'* in the various possible subcases. 
His well-known and practical ''method of exclusion" (arts. 319-322) employs 
successive small powers of primes as moduU. Another method (arts. 
327-8) is based on the theory of binary quadratic forms [cf. Smith^^°]. 
The congruence ax^-\-'bx-\-c=0 (mod m) is reduced (art. 152) to y^=}? — 
Aac (mod 4am). For each root y, it remains to solve 2ax-{-b=y (mod 4aw). 
Gauss^^^ showed in a somewhat incomplete posthumous paper that, if 
t is a prime and f~'^{t — l)=a''¥. . ., where a,h,. . . are distinct primes, the 
solution of a:"= 1 (mod t") may be made to depend upon the solution of a 
congruences of degree a, jS congruences of degree h, etc. Use is made of the 
periods formed of the primitive roots of the congruence, as in Gauss' theory 
of roots of unity. 
Legendre^^^ solved x^+a=0 (mod 2'") when a is of the form — 1 =F8a by 
i"Disquis. Arith., 1801, Arts. 60-65. 
"sWerke, 2, 1863, 199-211. Maser's German transl. of Gauss' Disq. Arith., etc., 1889, 589-601 
(comments, p. 683). 
"'Theorie des nombres, ed. 2, 1808, pp. 358-60 (Nos. 350-2). Maser, 2, 1893, 25-7. 
