Chap. IV] RESIDUE OF {U^^ — l)/^ MoDULO y. 107 
where ^l^-...^-^r = p, ij>0. The term given by r = p is a^. For p a 
prime, the left member is =a (mod p) and we have Fermat's theorem. By 
induction on r, 
Taking r = p, we have 
D. Mirimanoff^^ wrote ao for the least positive integer making aop+1 
divisible by the prime r<p, and denoted the quotient by r%i, where 6i is 
prime to r. Similarly, let a^ be the least positive integer such that a,p+6i = 
r%ij^i. We ultimately find an n for which 6n = l- Then 6„4.j = 6i. By (2), 
?b<~r-^i?'-+?6.>" - .^ 7r=gr2ei(modp). 
Oi »-0 Oj 
Let r belong to the exponent co modulo p and set 6w = p — 1. Then Sev=co, 
while 1, 61, . . . , 6„_i are the distinct residues of the eth powers of the integers 
< r and prime to r. Thus 
g^=eS -^ (mod p). 
The formula obtained by taking r a primitive root of p is included in the 
following, which holds also for any prime r : 
$,= 2 -* (mod p), 
ai being the least positive integer for which aip+iSi=0 (mod r). Set 
/3i = p— 5, p'p=l (mod r), 0<p'<r. Then ai=p'b — \ (mod r), 
]A;} being the least positive residue modulo r of k. Whence Sylvester's^ 
statement. 
J. S. Aladow^^ proved that (1) has at most (p=f1)/4 roots if p = 4?n=tl. 
A. Cunningham^^" listed 27 cases in which r^~^=l or r'=l (mod p'), 
r<p^~^, where Z is a divisor of p — 1. For the 11 cases of the first kind, 
p = 5, 7, 17, 19, 29, 37, 43, 71, 487. 
W. Fr. Meyer^^ proved by induction that, if p is a prime, x^~^ — l is 
divisible by p* {l^k<n), but not by p^'^^, for exactly p"~^~^ (p — 1)^ posi- 
tive integers a:<p" and prime to p, and is divisible by p" for the remaining 
p — 1 such integers. Set 
^=a+MiP+. ■ ■+MnP" (l^Q<P,0^iU><p), Xp=(a^'-a"' Vp 
"Jour, fiir Math., 115, 1895, 295-300. 
"St. Petersburg Math. Soc. (Rusaian), 1899, 40-44. 
"<»Messenger Math., 29, 1899-1900, 158. See Cuniungham"^, Ch. VI. 
"Archiv Math. Phys., (3), 2, 1901, 141-6. 
