218 History of the Theory of Numbers. [Chap, vii 
G. Picou^*^ applied to the case n = 2 Wronski's^^^ formula for the resi- 
dues of 71 th powers modulo M, M arbitrary. For example, if M = \Qa^\, 
(h=^Sa)-^ =pa{^h-iy (mod M). 
[If 8a were replaced by 4a, we would have an identity in h.] 
P. Bachmann^^ (pp. 344-351) discussed x"'=a (mod p"), p>2, p = 2. 
G. Arnoux-°^ solved x^^=79 (mod 3-5-7) by getting the residue 2 of 79 
modulo 7 and that of 14 modulo 0(7) =6 and solving x'^=2 (mod 7) by use 
of a table of residues of powers modulo 7. Similarly for moduli 3, 5. Take 
the product of the roots as usual. 
M. Cipolla-"'^ generahzed the results of Alagna-*^" to the case of a prime 
p = 2"'q-\-l, 7n>0, q an odd prime, including unity. For any divisor d of 
p — 1, the roots of x'^=N (mod p) are expressed as given powers of a primi- 
tive root a of p. If 2 belongs to the exponent 2''co modulo p, where w is 
odd, theng'= 1 (mod p) if and only if 2""^ is the highest power of 2 dividing m. 
Cunningham-"^" found the sum of the roots of (i/"=tl)/(?/±l) = 
(mod p). 
M. Cipolla'"^ proved the existence of an integer k such that k~ — q is a 
quadi'atic non-residue of the prime p not dividing the given integer q. Let 
Un = h^q\{kWqr-{k-Vqr\, 
v^=Viik+V¥^r+{k-V¥^r\. 
By expansion of the binomials it is shown that the roots of x^=q (mod p) 
are given by =*=W(p_i)/2 and by ±y(p+i)/2. These may be computed by use of 
Wn=2kWn_i—qWn-2 (mod p) {w = u or v), 
with the initial values Uq = 1, Wi = p; ^0 = 1? Vi = k. Although u^, y„ are the 
functions of Lucas, the exposition is here simple and independent of the 
theory of Lucas (Ch. XVII). 
M. Cipolla-°^ proved that if 5 is a quadratic residue and k^—q is a 
quadratic non-residue of an odd prime p, z~^q (mod p^) has the roots 
^lVq\{k+Vqr-{k-V~qr\, 
where r = p^~\p — 1)/2. Other expressions for the roots are 
^hq'{(k+V¥^r+{k-V¥^y\, 
t=ip^-2p^-' + l)/2, s = p^-\p + l)/2. 
Thus if Zi~=q (mod p), the roots modulo p^ are ^q'zi^^''^ (TonelH^^^). 
Finally, let n=TLpi^', where the p's are primes >3; take e, = =»=l when 
Pi=^l (mod 4). There exists a number A of the form k^—q such that 
^ML'iatermddiaire dea math., 8, 1901, 162. 
'^o* Assoc, frang. av. sc, 31, 1902, II, 185-201. 
»«Periodico di Mat., 18, 1903, 330-5. 
»'»«Math. Quest. Educ. Times, (2), 4, 1903, 115-6; 5, 1904, 80-1. 
"'Rendiconto Accad. Sc. Fis. e Mat. Napoli, (3), 9, 1903, 154-163. 
"8/6id., (3), 10, 1904, 144-150. 
