CHAP, vii] BINOMIAL CONGRUENCES. 219 (A/p1) = €1,. . ., (A/^)=€,, where the symbols are Legendre's. Call M the 1. c. m. of pi*~~I(pi—el)/2 for {=!,..., v. Then ^=g (mod ri) has the root A. Cunningham209 indicated how his tables may be used to solve directly ofe — 1 (mod p) for n = 2, 3, 4, 6, 12. From p = a2-f&2, we get the roots z==fc a/6 of #2= — 1 (mod p). Also p = a2+62 = c2-f-2d2 gives the roots =»=d(a+&)/(ce) and =*=c(a=*=&)/(2cte) of a;4= — 1 (mod p), where e = a orb. Again, p=A2+3£2 gives the roots (A-B)/(2£), (£-M)/(£-A), and their reciprocals, of ofel (mod p). M. Cipolla107 gave a report (in Peano's symbolism) on binomial congruences. M. Cipolla210 proved that if p is an odd prime not dividing q and if z2=g (mod p) is solvable, the roots are z= where Then x2=q (mod px) has the root 3pX~~V> e=(px-2px-1+l)/2. For p= (mod 4), x*=q (mod p) has the root 4 ly««-!- 2 y-^.3+2 s^««-i (z =£i^) * M. Cipolla211 extended the method of Legendre159 and proved that x2m=l+2*A (mod 2*), for A odd and s^ra+2, has a root where =J._ =(2m-l)(2-2m~l) . . .(^1-2OT~ TO> n are the coefficients in 0. Meissner212 gave for a prime p = 8n+5 the known root P+3 p-1 ? = D 8 of o;2=J[) (modp), D 4 =1 (modp). But if D(p-1)/4= -1 (mod p), a root is £](p-l)/2}!, since the square of the last factor is congruent to ( — l)(p+1)/2 by Wilson's theorem. Tamarkine and Friedmann213 expressed the roots of z*z=q (mod p) by a formula, equivalent to Cipolla's,210 209Quadratic Partitions, 1904, Introd., xvi-xvii. Math. Quest. Educ. Times, 6, 1904, 84-5; 7, 1905, 38-9; 8, 1905, 18-9. 310Rendiconto Accad. Sc. Fia. e Mat. Napoli, (3), 11, 1905, 13-19. au/Wd., 304-9. 21IArchiv Math. Phys. (3), 9, 1905, 96. "'Math. Aimalen, 62, 1906, 409.