Chap. VII] BiNOMIAL CONGRUENCES. 217 
0, 1,2,... until we reach a number of the form ^-\-z (found by extracting 
the square root). Then fc = s, so that the roots P-A^, P+l+Zc are found. 
N. Amici^^^ proved that if neither m nor 6 is divisible by the prime p, 
and if a is a given root of x'^=h (mod p), and if /3, g are (existing) integers 
such that 
i3(/)(p')-p'-i + l=mg, 
then of^'^"^ is a root of x™=6 (mod y^). Hence we limit attention to the 
case X = l. Consider henceforth x'^=h (mod p), where p = 2''/i+l is an 
odd prime, h being odd, and 6 not divisible by p. First, let ¥^8. Then 
6''=1 (mod p) is a necessary and sufficient condition for solvabihty and 
x= ± y^ are roots, where q is such that 2^q — 1 is divisible by /i. If gr is a 
quadratic non-residue of p, all 2" roots are given by ± 6''gr''', where e = 6i+2e2 
4- . . . +2*~^€^,_i, the ei taking the values and 1 independently. Finally, 
let A:<s. Then two roots ±(3 are determined by the method of TonelU, 
while all the roots are given by 
x==^^g'', t = e,+2e2+ . . . +2'-\_„ €^ = or 1. 
R. Alagna^°° considered a prime p = 4/c+l for which /b is a prime. Since 
2 is known to be a primitive root of p, it is easy to write down those powers 
of 2 which give all the roots of x'^=l (mod p), where d is one of the six 
divisors 2' or 2'k of p — 1, likewise of x'^^N, since N must be congruent to 
an even power of 2. For the modulus p^, we may apply the first theorem 
of Amici or proceed directly. The same questions are treated for a prime 
4A;+3 for which 2A; + 1 is a prime. 
A. Cunningham^"^ treated at length the solution of x^=\ (mod iV'), 
where iV is a prime, and gave tables showing all incongruent roots when 
< = 1, 2, N-^ 101, I any admissible divisor of iV — 1 ; also for a few additional 
f's when N is small. 
Cunningham^oi" treated a^= 1 (mod q^) and 3.2^= ± 1 (mod p). He^oi*- 
treated the problem to find 5''=+l or ±a, given a^=\, a''=^h (mod p), 
where ^ is odd and ^, x, 17 are the least values of their kind; also given 
a*=l, a'^^^h, a'=^c, to find the least /? and 7 such that h^=c, c^=6 
(mod p). 
W. H. Besant^°^ would solve y^ = ax+h by finding the roots s of s^=6 
(mod a). Then y = ar-\-s, x = ar^-\-2rs+{s'^ — h)/a. 
G. Speckmann^°^ replaced x"=A; (mod p) by the pair of congruences 
x"~^=r, xr^A; (mod p). In np+/c give to n the values 0, 1, 2, . . . until we 
find one for which np+k = rx such that, by trial, x"~^=r. The method is, of 
course, impractical. 
"'Rendiconti Circolo Mat. di Palermo, 11, 1897, 43-57. 
""Rendiconti Circolo Mat. di Palermo, 13, 1899, 99-129. 
"^Messenger of Math., 29, 1899-1900, 145-179. Errata, Cunmngham226, p. 155. See 13a of 
Ch. IV. 
"i^Math. Quest. Educ. Times, 71, 1899, 43-4; 75, 1901, 52-4. 
2"&/6td., (2), 1, 1902, 70-2. 
^o^Math. Gazette, 1, 1900, 130. 
'"'Archiv Math. Phys., (2), 17, 1900, 110-2, 120-1. 
