CHAP, xvij FACTOKS op a*=«=&*. 385 A. S. Bang33 discussed Ft(a) defined by (1). If p is a prime, Fj(a) has only prime factors ap*+l if d=opfc"1-l is prime to p, but has the factor p (and not p2) if d is divisible by p. Bang84 proved that, if a>l, t>2, Ft(a) has a prime factor at+1 except forFe(2). L. Gianni35 noted that if p is an odd prime dividing a—1 and jf divides ap—1, then p""1 divides a—I. L. Kronecker36 noted that, if Fn(z) is the function whose roots are the <t>(ri) primitive nth roots of unity, is an integral function involving only even powers of y. He investigated the prime factors q of Gn(x, s) for 8 given. If q is prime to n and s, then q is congruent modulo n to Jacobi's symbol (s/g). The same result was stated by Bauer.87 J. J. Sylvester38 called 0™— 1 the mth Fermatian function of 6. Sylvester39 stated that, for 0 an integer 5*1 or —1, 6 - *-— contains at least as many distinct prime divisors as m contains divisors >1 , except when 6= —2,m even, and 6=2, m a multiple of 6, in which two cases the number of prime divisors may be one less than in the general case. Sylvester40 called the above 0m a reduced Fermatian of index m. Ifm = np*, n not divisible by the odd prime p, 0TO is divisible by ptt, but not by p*44, if 6 - 1 is divisible by p. If m is odd and 6 — 1 is divisible by each prime factor of m, then 0m is divisible by m and the quotient is prime to m. Sylvester40" stated that if P=l+p+. . -+Pr~1 is divisible by g, and p, r are prunes, either r divides g — 1 or r=q divides p— 1. If P=gJ and p, r, j are primes, j is a divisor of g— r. R. W. Genese easily proved the first statement and W. S. Foster the second. T. Pepin41 factored various on— 1, including a = 79, 67, 43, n = 5; a =7, n = ll; a=3, n = 23; a = 5 or 7, n = 13 (certain ones not in the tables by Bickmore49). H. Scheffler42 discussed the factorization of 2r+l by writing possible factors to the base 2, as had Beguelin.8 He noted (p. 151) that, if m = 2n~1, 1 +2(2w+1)n = (1 +2n)2 { 1 -2m+ (2m - l)2n - (2m - 2)22n + . . . — 2-2(2m"2)n+2(2m""1)rl}. His formula (top p. 156), in which 2rl"~1 is a misprint for 22*"1, is equivalent to that of LeLasseur.23 "Tidsskrift for Mat., (5), 4, 1886, 70-80. M/6id., 130-137. "Periodico di Mat., 2, 1887, 114. '•Berlin Berichte, 1888, 417; Werke, 3, 1, 281-292. 87Jour. ftir Math., 131, 1906, 265-7. '"Nature, 37, 1888, 152. «/Wd., p. 418; Coll. Papers, 4, 1912, 628. *°Comptes Rendue Paris, 106, 1888, 446; Coll. Papers, 4,. 607. 40aMath. Quest. Educ. Times, 49, 1888, 54, 69. 41Atti Accad. Pont. Nuovi Lincei, 49, 1890, 163. Cf . Escott, Messenger Math., 33, 1903-4, 49. «iii> TxatilaYitViork™ 1BQ1 1J.7-.17ft