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CHAP, vii]                       BINOMIAL CONGRUENCES.                                 219
(A/p1) = 1,. . ., (A/^)=,, where the symbols are Legendre's. Call M the 1. c. m. of pi*~~I(piel)/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+32 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.