218 HlSTOKY OF THE THEORY OF NtJMBEKS. [CHAP. VII G. Picou204 applied to the case n=2 Wronski's169 formula for the residues of nth powers modulo M, M arbitrary. For example, if M = 16a=fcl, (/i=b8a)2=^a(4/i-l)2 (mod M). [If 8a were replaced by 4a, we would have an identity in h.] P. Bachmann104 (pp. 344-351) discussed xmz=a (mod pa), p>2, p = 2. G. Arnoux205 solved x14=79 (mod 3-5-7) by getting the residue 2 of 79 modulo 7 and that of 14 modulo <£(7) = 6 and solving #2=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. Cipolla206 generalized the results of Alagna200 to the case of a prime p = 2mq+l, m>Q, q an odd prime, including unity. For any divisor d of p—l, the roots of xds=N (mod p) are expressed as given powers of a primitive root a of p. If 2 belongs to the exponent 2pco modulo p, where co is odd, theng*= 1 (mod p) if and only if 2""1 is the highest power of 2 dividing m. Cunningham2060 found the sum of the roots of (?/w=*=l)/(7/=fc 1)^=0 (mod p). M. Cipolla207 proved the existence of an integer k such that k2— q is a quadratic non-residue of the prime p not dividing the given integer q. Let By expansion of the binomials it is shown that the roots of x2=zq (mod p) are given by ^=u(p^l}/2 and by ^v(p+l}/2. These may be computed by use of wn= 2ton_1 — qwn-.2 (mod p) (w=u or v) , with the initial values w0 = l, % = p; vQ = l, vl = k. Although un, vn are the functions of Lucas, the exposition is here simple and independent of the theory of Lucas (Ch. XVII). M. Cipolla208 proved that if q is a quadratic residue and k2— q is a quadratic non-residue of an odd prime p, z2=q (mod px) has the roots where r = p*~l(p — 1)/2. Other expressions for the roots are Thus if zfzsq (mod p), the roots modulo px are ^V*""1 (Tonelli193). Finally, let n^TLp*, where the p's are primes >3; take ^ = ±1 when ^= =F! (mod 4). There exists a number A of the form k2 — q such that 2ML1interm6diaire des math., 8, 1901, 162. 206Assoc. frang. av. sc., 31, 1902, II, 185-201. 206Periodico di Mat., 18, 1903, 330-5. st)5aMath. Quest. Educ. Times, (2), 4, 1903, 115-6; 5, 1904, 80-1. 207Rendiconto Accad. Sc. Fis. e Mat. Napoli, (3), 9, 1903, 154-163. S08/6id., (3), 10, 1904, 144-150.