220 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. VII (p-3)/2 For, according as y2 is or is not =# (mod p), we have y \ 1 — (i/2—q)p~* [ —y or 0 (mod p). We can express s2m-H in terms of Bernoullian numbers. A. Cunningham214 gave a tentative method of solving x2=a (mod p). He214a noted that a root F=2*?2 of F4== -1 leads to the roots of s/8=-l (mod p). M. Cipolla215 employed an odd prime p and a divisor n of p — l=nv. If TI,. .., rv form a set of residues of p whose nth powers are incongruent, and if g"=l (mod p), then xnz=q (mod p) has the root £s= 2J Akqk, Ak= — nS r/1*"1. For n=2, this becomes his210 earlier formula by taking 1,2,..., (p —1)/2 as the rjs. Next, let p—l=mju, where m and JLI are relatively prime and m is a multiple of n. If 7 and 6 belong to the exponents m and ^ modulo p, the products 7r5* (r<m/n, $</*) may be taken as r1?..., ry. According as or not (mod ju), we have AA= — n/A^—-pi—:— or AA=0 (mod p). If n is a prime and nr is its highest power dividing p — 1, there exists a number o> not an nth power modulo p and we may set m = nr, 7=coM (mod p) . In particular, if n=2, afeg has the root where w is a quadratic non-residue of p. If p=5 (mod 8), we may take co = 2 and get P+3 - M. Cipolla216 considered the congruence, with p an odd prime, zpf==a (mod pw), r<m, a necessary condition for which is that h = (a?T—a)/pr+1 be an integer. Determine A by c/As/i (mod pm). Then the given congruence has the root axQ if z0 is a root of This is proved to have the root »*Math. Quest. Educ. Times, (2), 13, 1908, 19-20. S14a/Wd., 10, 1906, 52-3. JlsMath. Annalen, 63, 1907, 54-61. s»Atti R. Accad. Lincei, Rendiconti, (5), 16, 1, 1907, 603-8.