Chap. VII] BiNOMIAL CONGRUENCES. 221 
where Ci = l/p*", . . . are given by the expansion 
^1—2 ==l—CiZ — C2^ — 
M. Cipolla^" treated x"=a (mod p"*) where n divides 4>{p^). We may 
set n = p'v, where v divides p — 1. Determine integers a, j3 such that 
Then the initial congruence has the root yx^" if y^^=a^ (mod p"*), solved as 
in his preceding paper, and if Xi is a root of x''=a (mod p"*). The latter 
has the root 
^ A;=0 t = l 
where t={p — l)/v, pi^rf""' (mod p'"), ri, . . ., r^ being integers prime to p 
such that their j/th powers are incongruent and form a group modulo p"*. 
K. A. Posse^^^ gave a simplified exposition of Korkine's^^^ method of 
solving binomial congruences. Cf. Posse/^^ Schuh.^^^'^ 
F. Stasi^^^ proved that we obtain all solutions of x^=a^ (mod n), where 
n is odd and prime to a, by expressing n as a product of two relatively 
prime factors P and Q in all ways, setting x — a = Pz and finding z from 
Pz+2a=0 (mod Q). [Instead of his very long proof, it may be shown at 
once that we may take x — a, x+a divisible by P, Q, respectively.] 
L. Grosschmid^^° gave for the incongruent roots of x^^r (mod M) an 
expHcit formula obtained by means of the ideal factors of ilf in a quadratic 
number-field. 
L. Grosschmid^^^ treated the roots of quadratic binomial congruences. 
A. Cunningham^^^ solved x^= —1 (mod p), where p = 616318177 is a prime 
factor of 2^^ — 1; by using various small moduli, he obtained p = 24561^ + 
36161 
L. von Schrutka^^^'' used a correspondence between the integers and 
certain rational numbers to treat quadratic congruences without novelty as 
to results. The method will be given under the topic Fields in a later 
volume of this History. 
Grosschmid^^^ employed the products R and N of all the quadratic 
residues and non-residues, respectively, ^2n of a prime p = 4n+l. Then 
R^={-iy+\ iv2=(_i)« (modp). 
2"Atti R. Accad. Lincei, Rendiconti, (5), 16, I, 1907, 732-741. 
"'Charlkov Soobsc. Mat. Obs6 (Report Math.Soc. Charkov), (2), 11, 1910, 249-268 (Russian). 
"»I1 BoU. Matematica Gior. Sc.-Didat., 9, 1910, 296-300. 
«20Jour. fur Math., 139, 1911, 101-5. 
""Math. 6s Phys. Lapok, Budapest, 20, 1911, 47-72 (Hungarian). 
222Math. Questions Educat. Times, (2), 20, 1911, 33-4 (76). 
2««Monatshefte Math. Phys., 23, 1912, 92-105. 
223Archiv Math. Phys., (3), 21, 1913, 363; 23, 1914-5, 187-8. 
