206 History of the Theory of Numbers. [Chap, vii 
Since In'—qp' = 1, the first gives 5^"'= 1. Hence 
Conversely, if B^'=l, 
a^p-i_l=a^p'-_j5p'/ (mod p) 
has the factor x" — B\ so that (Lagrange^) congruence (2) has co roots. 
If 4n divides p — 1, the roots of x^"= —1 (mod p) are the odd powers of 
an integer belonging to the exponent 4n modulo p. 
Let n divide p — l, and 7n divide {p — l)/n. Let co be the g. c. d. of 
m , n and set n = cov. Determine positive integers I and q such that lv — qm = l. 
If 5'"= =•= 1 (mod p), (1) is satisfied by the roots of x'^^B'y (mod p), where 
y ranges over the roots of ?/"= (=t 1)^ (mod p). For, the last two congruences 
give 
x'' = x'"'=B''^y''=B'"^+\=i=iy=B (mod p). 
Hence by means of the roots of ?/''=±l, we reduce the solution of (1) to 
binomial congruences of lower degrees. In particular, let n = 2, m = (p — 1)/2, 
and let 2 be prune to ?«, so that p = 4:a — l,l = a,q = l. Then x^ = B (mod p) 
requires that 5"* = 1 , so that we have the solutions .t = =*= 5" without trial 
(Lagrange^^^). Next, if n = 2 and 5^'^+^= —1, the theorem gives x = B'''^^y, 
where ?/- = — L But we may generalize the last result. Consider x" + c^ = 
(mod p). Since p must have the form 4a+l, we have p=f^-\-g^. Deter- 
mine u and z so that c = gu—pz. Then x = fu (mod p). 
Let a belong to the exponent nw modulo p, where w divides (p — l)/n. 
Then the roots of B"' = l (mod p) are B = a"*' (m = 1,- • •, w) — 1), and, for a 
fixed B, the roots of (1) are x = 0'"""+" (m = 0, 1, . . . , n - 1). For, a" belongs 
to the exponent w, whence B = a'"'. 
Legendre^^^ gave the same theorems in his text. He added that, know- 
ing a root 6 of (1), it is easy to find a root of x" = B (mod p"), with the 
possible exception of the case in which n is divisible by p. Let6"—B = Mp 
and set a:=^+i4p. Then a;" — 5 is divisible by p^ if 
M+'nB''-^A=pM', 
which can be satisfied by integers A, Af' if n is not divisible by p. To solve 
(1) when p is composite, p = a°6^ . . . , where a, 6, . . . are distinct primes, deter- 
mine all the roots X of X" = B (mod a°), all the roots ^ of ijl" = B (mod b^), 
Then if x=\ (mod a°), x=iJi (mod 6^),. . ., x will range over all the roots 
of (1). 
Legendre^^^ noted that if p is a prime 8n+5 we can give explicitly the 
solutions of a:^+a = (mod p) when it is solvable, viz., when a'*"''"^ = 1. For, 
either «-"+' + 1=0 and x = a"+' is a solution, or a-"+^-l=0 and (9 = a"+^ 
satisfies d^ — a^O (mod p), so that it remains only to solve x^-\-d' = 0, which 
was done at the end of his^^* memoir. For p = 8?i+l, let n = a^, where a 
is a power of 2 and /3 is odd; if 0"= ± 1, x^-\-a = can be solved as in the 
i"Th(5orie des nombres, 1798, 411-8; ed. 2, 1808, 349-357; ed. 3, 1830, Nos. 339-351; German 
transl. by Maser, 1893, 2, pp. 15-22. 
^"Ibid., 231-8; ed. 2, 1808, pp. 211-219; Maser, I, pp. 246-7. 
