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 la 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.