Chap. VII] BiNOMIAL CONGRUENCES. 211 
x-h={-iy-'P,-u r^{-iy-\dh+irP^.^ (mod p). 
By taking ^=(1^/')^ and replacing 1 by (-1)^+^ in 0(a;-/i) = l, the last 
results become the fundamental formula given without proof by Wronski^^^ 
in his Reforme des Mathematiques. 
G. L. Dirichlet^^^ discussed the solution of x^^D for any modulus. 
G. F. Meyer"^ gave an elementary discussion of the solution of x^=b 
(mod k), for k a prime, power of prime, or any integer. 
V. A. Lebesgue^^^ employed a prime p, a divisor n of p — l=nn', and a 
number a belonging to the exponent n' modulo p. Then the roots of 
a;" = a (mod p) are a"6^, where h is not in the period of a, and 6 is a quadratic 
non-residue of p if a is a quadratic residue, and 6" is the least power of h 
congruent to a term of the period of a. If we set 6" = a" (mod p), then 
must na-\-v^ = l (mod n'). The roots x are primitive roots of p. In the 
construction of a table of indices, his method is to seek a primitive root 
giving to ±2 the minimum index (rather than to ±10, used by Jacobi); 
thus we use the theorem for a= ±2. 
Lebesgue^'^^ gave reasons why the conditions imposed on h in his pre- 
ceding paper are necessary. He added that when we have found that 
x" = a (mod p) leads to a primitive root x = g oi p,\i is easy to solve x'"=r 
(mod p) when m divides p — 1, by expressing r as a power of g by the equiva- 
lent of an abridged table of indices. 
Lebesgue^^*^ noted that the usual method of solution by indices leads 
to the theorem: If a belongs to the exponent e modulo p, and if n divides 
p — 1, and we set n = e'm, where e' has only prime factors which divide e, 
while m is prime to e, then, for every divisor M of m, x'^^a (mod p) has 
e'(j){M) roots belonging to the exponent M. 
If a belongs to the exponent e modulo p, there are e0(n) numbers h, not 
in the period of a, for which 6"= a' (mod p), with n sl minimum. A common 
divisor of n and i does not divide e. Then the n roots of x"= a (mod p) are 
a'6", where nt — iu — l = ev, t<e, u<n. This generahzation of his^^'' earlier 
theorem is used to find the period of a primitive root of p from the period of 2. 
R. Gorgas"^ stated that, if p is the residue modulo M of the pth term of 
](M-l)/2[^. . .,2^ 1^, then p(p-l)=p±m+ikf a, according as ilf = 4m=t:l. 
Take the lower signs and solve for p ; we get 
2p = l±6, 62 = M(4a-l)+4p. 
Set 4p = Mc+p'. Hence the initial equation x^ = My+p has been replaced 
by 6^ = M(4a-fc — l)+p' of like form. Let p' be the p'th place from the 
end. The process may be repeated until we reach an equation P(P — 1) 
= MA-\-p^—m solvable by inspection. 
"^Zahlentheorie, 1863, §§32-7; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. 
»"Archiv Math. Phys., 43, 1865, 413-36. 
"^Comptes Rendus Paris, 61, 1865, 1041-4. 
"'Ihid., 62, 1866, 20-23. 
"«/6id., 63, 1866, 1100-3. 
"^Ueber Losung dioph. Gl. 2. Gr., Progr., Magdeburg, 1867. 
