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220                         HlSTOKY OF THE THEORY OF NUMBERS.               [CHAP. VII
For, according as y2 is or is not =# (mod p), we have y \ 1  (i/2q)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 pl=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?Ta)/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.
sAtti R. Accad. Lincei, Rendiconti, (5), 16, 1, 1907, 603-8.