Chap. XVI] FACTORS OF a" ±6". 385 
A. S. Bang^^ discussed F<(a) defined by (1). If p is a prime, F^i^{a) has 
only prime factors ap*+l if c? = a^ ~ —1 is prime to p, but has the factor p 
(and not p^) if d is divisible by p. 
Bang^^ proved that, if a>l, t>2, Ft{a) has a prime factor a^+1 except 
forF6(2). 
L. Gianni^^ noted that if p is an odd prime dividing a — 1 and p" divides 
aF — \, then p""^ divides a — 1. 
L. Kronecker^^ noted that, if F^iz) is the function whose roots are the 
<^(n) primitive nth roots of unity, 
{x-yY^-^F^{^^=GAx,y') 
is an integral function involving only even powers of y. He investigated the 
prime factors q of Gn{x, s) for s given. If q is prime to n and s, then q is 
congruent modulo n to Jacobi's symbol {s/q). The same result was stated 
by Bauer.^^ 
J. J. Sylvester^^ called ^"" — 1 the mth Fermatian function of 6. 
Sylvester^^ stated that, for 6 an integer 9^1 or —1, 
ft =?JZi 
contains at least as many distinct prime divisors as m contains divisors > 1 , 
except when 0= — 2, m even, and ^ = 2, m a multiple of 6, in which two cases 
the number of prime divisors may be one less than in the general case. 
Sylvester^" called the above 6^ a reduced Fermatian of injdex m. lim = np", 
n not divisible by the odd prime p, 6^ is divisible by p", but not by p"'^^, if 
— 1 is divisible by p. If m is odd and ^ — 1 is divisible by each prime factor 
of m, then dm is divisible by m and the quotient is prime to m. 
Sylvester^^" stated that if P=l+p+ . . .+p''~^ is divisible by q, and 
p, r are primes, either r divides q — 1 or r=q divides p — l. li P=q^ and 
p, r, j are primes, j is a divisor of q—r. R. W. Genese easily proved the 
first statement and W. S. Foster the second. 
T. Pepin^^ factored various a" — 1, including a = 79, 67, 43, n = 5; a = 7, 
n = ll', a = S, 71 = 23; a = 5 or 7, w = 13 (certain ones not in the tables by 
Bickmore^^) . 
H. Scheffler*^ discussed the factorization of 2'"+! by writing possible 
factors to the base 2, as had Beguelin.^ He noted (p. 151) that, if m = 2"~\ 
1^2(2-+!'" = (i+2")2{i_2m+(2m-l)2"-(2m-2)22" 
I o o(2m— 2)n_|_o(2m— l)n) 
His formula (top p. 156), in which 2^*"^ is a misprint for 2^''"^ is equivalent 
to that of LeLasseur.^^ 
"Tidsskrift for Mat., (5), 4, 1886, 70-80. ^*Ibid., 130-137. sspgriodico di Mat., 2, 1887, 114. 
36Berlin Berichte, 1888, 417; Werke, 3, I, 281-292. "jour, fiir Math., 131, 1906, 265-7. 
"Nature, 37, 1888, 152. ^Ubid., p. 418; CoU. Papers, 4, 1912, 628. 
"Comptes Rendus Paris, 106, 1888, 446; CoU. Papers, 4, 607. 
"«Math. Quest. Educ. Times, 49, 1888, 54, 69. 
«Atti Accad. Pont. Nuovi Lincei, 49, 1890, 163. Cf. Escott, Messenger Math., 33, 1903-4, 49. 
«Beitrage zur Zahlentheorie, 1891, 147-178. 
