I 
Chap. VII] BiNOMIAL CONGRUENCES. 219 
(A/pi) = €i,. . ., (A/pJ = e^, where the symbols are Legendre's. Call M 
the 1. c. m. of pl^~\pi — ei)/2 for {=1,..., v. Then z^=q (mod n) has 
the root 
A. Cunningham^°^ indicated how his tables may be used to solve 
directly x''= —1 (mod p) for n = 2, 3, 4, 6, 12. From p = a^-\-h^, we get the 
roots re = ± a/6 of a;^=— 1 (modp). Also p = a^+ 6^ = c^+2(i^ gives the roots 
=i= d{a-\-b) / (ce) and ±c(a±6)/(2de) of x^^—1 (mod p), where e = a or 6. 
Again, p = A^+SB^ gives the roots iA-B)/{2B), {B+A)/{B-A), and 
their reciprocals, of x^=l (mod p). 
M. Cipolla^°^ gave a report (in Peano's symbolism) on binomial con- 
gruences. 
M. Cipolla^^° proved that if p is an odd prime not dividing q and if 
z^=q (mod p) is solvable, the roots are 
z= ^2{qs,+q\+q\+ . . . +5'^~'%-4+Sp-2) 
where 
s,=r+z+...+ 
m- 
Then x^=q (mod p'^) has the root z^^ V, e = (p^-2p^-^ + l)/2. For p=l 
(mod 4), x'^=q (mod p) has the root 
4 1 5^S2,_i- 2 q^-%j_s+2 S g^s^^.i (z = ^) • 
1=1 y=i i=l \ 1 / 
M. Cipolla^^^ extended the method of Legendre^^® and proved that 
x^"'=l-\-TA (mod 2*), 
for A odd and s^w+2, has a root 
x = l+2^Aci-22M2^2+- • • + (-l)"'"'2"^A"c„, n=r _^~^ J , 
where ^_]_ ^(2'"-l)(2-2'"-l) . . .(n^l-2'"-l) 
are the coefficients in 
il+zy^'"' = l+c,z-c,z'+c,z'- . . . -(-1)X2"+. . .. 
0. Meissner^^^ gave for a prime p = Sn+5 the known root 
£+3 £-1 
^ = D » oix^=D (modp), D * =i (modp). 
But if 2)^p-i)/4= _i (mod p), a root is ^](p — 1)/2)!, since the square of the 
last factor is congruent to ( — l)(p+^)/2 j-^y wrjigon's theorem. 
Tamarkine and Friedmann^^^ expressed the roots of z^^q (mod p) by a 
formula, equivalent to Cipolla's,^^° 
^osQuadratic Partitions, 1904, Introd., xvi-xvii. Math. Quest. Educ. Times, 6, 1904, 84-5; 7, 
1905, 38-9; 8, 1905, 18-9. 
""Rendiconto Accad. Sc. Fis. e Mat. Napoli, (3), 11, 1905, 13-19. 
"i/6id., 304-9. 
'"Archiv Math. Phys. (3), 9, 1905, 96. 
2"Math. Annalen, 62, 1906, 409. 
