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CHAP, vii]                       BINOMIAL CONGRUENCES.                                 221
where cx = l/pr, ... are given by the expansion £l— 2 =1— d2— C222--
M. Cipolla217 treated rcnsa (mod pm) where n divides <£(pm)-   We may set n=pV, where v divides p— 1.   Determine integers a, 0 such that
apr+v!3= 1 {mod p1"-^?- 1) [ .
Then the initial congruence has the root yxf if yprz=a? (mod pm), solved as in his preceding paper, and if xl is a root of x"=a (mod pm). The latter has the root
where £=(p — l)/p, p.-sr?1"'1 (mod pm), r1?..., rt being integers prime to p such that their *>th powers are incongruent and form a group modulo pm.
K. A. Posse218 gave a simplified exposition of KorkineV18 method of solving binomial congruences. Cf. Posse,129 Schuh.123"4
F. Stasi219 proved that we obtain all solutions of x2=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. Grosschmid220 gave for the incongruent roots of ofer (mod M) an explicit formula obtained by means of the ideal factors of M in a quadratic number-field.
L. Grosschmid221 treated the roots of quadratic binomial congruences.
A.Cunningham222 solved xz= — 1 (mod p), where p = 616318177 is a prime factor of 237—1; by using various small moduli, he obtained p = 245612-f-36162.
L. von Schrutka2220 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.
Grosschmid223 employed the products R and N of all the quadratic residues and non-residues, respectively, ^2n of a prime p = 4n+l. Then
#2=(-l)n+1,          Ar2s( —l)n (mod p).
»7Atti R. Accad. Lincei, Rendiconti, (5), 16, I, 1907, 732-741.
""Charlkov Soob§6. Mat. Ob§6 (Report Math.Soc. Charkov), (2), 11,1910, 249-268 (Russian).
»'I1 Boll. Matematica Gior. Sc.-Didat., 9, 1910, 296-300.
»°Jour. 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).
2J2«Monatshefte Math. Phys., 23, 1912, 92-105.
M»Archiv Math. Phys., (3), 21, 1913, 363; 23, 1914-5, 187-8.