220 History of the Theory of Numbers. [Chap, vii 
(p-3)/2 
z==b2 2 q^''-''-"'s2^+,. 
m = 
For, according as 2/^ is or is not =q (mod p), we have 
y\i-{y^-qy~'\=y or o (mod p). 
We can express S2m+\ in terms of Bemoullian numbers. 
A. Cunningham-^^ gave a tentative method of solving x'^=a (mod p). 
He-^^^" noted that a root Y=2r]^ of Y^=-l leads to the roots of y^=-l 
(mod p). 
M. Cipolla^^^ employed an odd prime p and a divisor n of p — l=ni/. 
If Ti, . . ., r, form a set of residues of p whose nth powers are incongruent, 
and if ^'=1 (mod p), then x''=q (mod p) has the root 
k=0 ;=1 
Forn = 2, this becomes his^^° earUer formula by taking 1,2,..., (p — 1)/2 as 
the r's. Next, let p — l=mji, where m and /x are relatively prime and m 
is a multiple of n. If 7 and 8 belong to the exponents m and /x modulo p, 
the products 7''5* {r<m/n, s<iJi) may be taken as ri,. . ., r,. According as 
nk= 1 or not (mod /x), we have 
y{nk-l)m/n -j^ 
At= -ufi — „^_i - or Ak=0 (mod p). 
7 — i 
If n is a prime and n" is its highest power dividing p — 1, there exists a 
number co not an nth power modulo p and we may set m = n'', 7=0)" (mod p). 
In particular, if n = 2, x^^q has the root 
_ 1 P+2^-1 2'"-'-l 
2 «=o 
where co is a quadratic non-residue of p. If p=5 (mod 8), we may take 
CO = 2 and get 
M. CipoUa^^® considered the congruence, with p an odd prime, 
x^ =a (mod p"*), r<7n, 
a necessary condition for which is that h = {a^ — a)/p'"^' be an integer. 
Determine A by a^ A=h (mod p"*). Then the given congruence has the 
root axo if Xq is a root of 
:r'''=l-^p'+' (modp"*). 
This is proved to have the root 
'"Math. Quest. Educ. Times, (2), 13, 1908, 19-20. 
^^"^Ibid., 10, 1906, 52-3. 
"»Math. Annalen, 63, 1907, 54-61. 
>"Atti R. Accad. Lincei, Rendiconti, (5), 16, I, 1907, 603-8. 
