Chap. VII] BiNOMIAL CONGRUENCES. 205 
as those of the term t/'""-" for ?/ = !,..., mn, and hence equal {mn — n)\, 
so that Q(y) is not divisible by p for some values 1, . . . , mn of y. 
Euler^^^ recurred to the subject. The main conclusion here and from 
his former paper is the criterion that, if p = mn-]-l is a prime, x''=a (mod p) 
has exactly n roots or no root, according as a"^=l (mod p) or not. In 
particular, there are just m roots of a""^!, and each root a is a residue of 
an nth power. 
Euler^^^" stated that, if aq-\-h=p'^, all the values of x making ax-\-b b, 
square are given by a;= ay'^^2py-\-q. 
J. L. Lagrange^^^ gave the criterion of Euler, and noted that if p is a 
prime 4n+3, B'-^'^^^^ — l is divisible by p, so that x=B'''^^ is a root of 
x^=B (mod p). Given a root ^ of the latter, where now p is any odd prime 
not dividing B, we can find a root of x^=B (mod p^) by setting x = ^-\-\p, 
i^-B = poi. Then x^-B = {\^-{-n)p'^ if 2|X+co=MP. The latter can be 
satisfied by integers X, jjl since 2^ and p are relatively prime. We can pro- 
ceed similarly and solve x^=B (mod p"). 
Next, consider ^^=B (mod 2"), for n>2 and B odd (since the case B 
even reduces to the former). Then ^ = 2z-\-l, ^^ — B = Z-\-\—B, where 
Z = 4:z{z-\-l) is a multiple of 8. Thus 1—B must be a multiple of 8. Let 
w>3 and 1-B = 2'^,r>3. If r^n, it suflaces to take 2 = 2""^, where f is 
arbitrary. If r<n, Z must be divisible by 2'', whence 2 = 2'""^^ or 2*""^^ — 1. 
Hence w=^{2'-^i:=i=l)-\-p must be divisible by 2"-''. If n-r^r-2, it 
suffices to take f =^iS divisible by 2""''. The latter is a necessary condition 
if n-r>r-2. Thus ^ = 2'-^p=F^, w = 2'-\f=t=p). Hence f ±p must be 
divisible by 2""^'^+^. We have two sub-cases according as the exponent of 
2 is ^ or >r — 1; etc. 
Finally, the solution of x^=B{mod m) reduces to the case of the powers of 
primes dividing m. For, if / and g are relatively prime and ^^ — Bis divisible 
by /, and \p^—B by g, then x^—B\b divisible by fg ii x= jif^ ^ = vg^\l/. But 
the final equality can be satisfied by integers /z, v since / is prime to g. 
A. M. Legendre^^^ proved that if p is a prime and co is the g. c. d. of n 
and p — \ = oip', there is no integral root of 
(1) a:"=j5(modp) 
unless B^'= 1 (mod p) ; if the last condition is satisfied, there are co roots 
of (1) and they satisfy 
(2) x'^^B^ (mod p), 
where I is the least positive integer for which 
(3) ln — q{p — \)=o}. 
For, from (1) and x''-^=l, we get x^''=B\ x^^^-^^^l, and hence (2), by use 
of (3). Set n = oin'. Then, by (2) and (1), 
^n'l^^n^2, 5P''=a;P'"=<rP-l=l (mod p) . 
"2Novi Comm..Petrop., 8, 1760-1, 74; Opusc. Anal. 1, 1772, 121; Comm. Arith., 1, 274, 487. 
"^aOpera postuma, I, 1862, 213-4 (about 1771). 
"^Mem. Acad. R. Sc. Berlin, 23, ann6e 1767, 1769; Oeuvres, 2, 497-504. 
"^M6m. Ac. R. Sc. Paris, 1785, 468, 476-481. (Cf. Legendre.i^^) 
