108 HISTORY OF THE THEOBY OF NUMBEKS. [CHAP.IV If k is the least index for which /iftf^Xfc, M*—^ (mod p) for h<k} then A1""1 — ! is divisible by p*, but not by pl+k. A. Palmstrom and A. Pollak15 proved that, if p is a prime and n, m are the exponents to which a belongs modulo py p2, respectively, then anp— 1 is divisible by p2, so that w is a multiple of n and a divisor of np, whence m=n or pn. Thus according as a?~~l is or is not =1 (mod p2), m=n or m=np. Worms de Romilly150 noted that, if w is a primitive root of p2, the incon-gruent roots of x*~lz=l (mod p2) are co;p(j = l, . . ., p-~l). J. W. L. Glaisher16 proved that if r is a positive integer <p, p a prime, where gn is the sum of the nth powers of 1 r-1 ,1 2 r-1 'f r+[(r~l)cr]; 2r+cr'' ' "' (r being the least positive residue modulo r of —p. If /*>• is the least positive solution of ar/jii^i (mod r), viz., pjJLi+i=Q, then gi^+^+ f m . +^=L+ J^+ J^+ mmt+ ^-i | 0i + Set Mr = 0, fjLi+3r =Mi- Then - 2,=0 (mod p). Sylvester's corrected results are proved. From (l+l)p, For r'=r+kp, let>/ be the positive root of pju/+^=0 (mod r'). Then r">-i = l+felp+KV-Wp It is shown that, for some integer t, k k2 2^--j5 (mod ^+3)< k k k2 2k k2 - = *p, ^-02s-2^--j5+2«s-^l--s (mod p), Glaisher,17 using the same notations, gave r-^l+p(^+|+;..+fcl) (modp'). "L'interm^diaire des math., 8, 1901, 122, 205-6 (7, 1900, 357). 1Bfljm, 214-5. 16Quar. Jour. Math,, 32, 1901, 1-27, 240-251. "Messenger Math,, 30, 1900-1, 78.