386 History of the Theory of Numbers. [Chap, xvi 
E. Lucas*^ gave algebraic factors of 
K. Zsigmondy^ proved the existence of a prime dividing a'^ — h'^, but no 
similar binomial with a lower exponent, exceptions apart (cf . Bang,^^- ^ 
Birkhoff^-). 
J. W. L. Glaisher*^ gave the prime factors of p^ — ( — l)^?-^)/^ ^^^ ^^^^i 
prime p<100. 
T. Pepin^^ proved that (31'-l)/30, (83'-l)/82, (2*^ + l)/(3-83) are 
primes. 
A. A. Markoff'^^ investigated the greatest prime factor of n^+1. 
W. P. Workman^^ noted the factors of 3^*+Hl [due to Catalan^^] 
and 2^"* + l, and stated that Lucas^^ (p. 326) gave erroneous factors of 2'^^+l. 
C. E. Bickmore*^ gave factors of a"-l for n^50, a = 2, 3, 5, 6, 7, 10, 
11, 12. 
Several^^" proved that n" — 1 is divisible by 4n+l if 47i+l is prime. 
A. Cunningham^° gave 43 primes exceeding 9 milUon which are factors _ 
of (x5±l)/(x±l), and factors of 3'°+l, 3''-l, S^^+l, S'^^'+l, 5'^-l, ] 
5^Hl, 5^'-l, S^^+l, 53^-1. 
A. Cunningham^^ considered at length the factorization of Aurifeuillians, 
i. e., the algebraically irreducible factors of 
n+l 
{n,7?Y^+ {2n^fT, (nix2)«+ ( - \)~{n^'\?r {n,n^ = n), 
where n^ and x are relatively prime to ^2 and ?/, while n has no square factor, 
and is odd in the second case. Aurifeuille had found them to be expressible 
algebraically in the form P^ — Q'^. There are given factors of 2"+! for 
n even and ^102, and for n = 110, 114, 126, 130, 138, 150, 210. 
A. Cunningham^^ factored numbers a"=*= 1 by use of tables, complete to 
p = 101, giving the lengths I of the periods of primes p and their powers 
< 10000 to various bases q, so that q^= 1 (mod p or p^). 
A. Cunningham and H. J. WoodalP^ gave factors of A^ = 2''10"±l for 
x^30, a^ 10, and for further sets; also, for each prime p^3001, the least 
a and the least corresponding x for which p is a divisor of N. Bickmore 
(p. 95) gave the linear and quadratic forms of factors of A^. 
T. Pepin^ factored a^-1 for a = 37, 41, 79; also^^ 151^-1. 
"Th^orie des nombres, 1891, 132, exs. 2-4. 
"Monatshefte Math. Phys., 3, 1892, 283. Details in Ch. VII, Zsigmondy." 
«Quar. Jour. Math., 26, 1893, 47. 
*«Memorie Accad. Pont. Nuovi Lincei, 9, I, 1893, 47-76. 
♦^Comptea Rendus Paris, 120, 1895, 1032. "Messenger Math., 24, 1895, 67. 
"/6id., 25, 1896, 1-44; 26, 1897, 1-38; French transl., Sphinx-Oedipe, 7, 1912, 129-44, 155-9. 
"^Math. Quest. Educ. Times, 65, 1896, 78; (2), 8, 1905, 97. 
»»Proc. London Math. Soc., 28, 1897, 377, 379. "/bid., 29, 1898, 381-438. 
"Messenger Math., 29, 1899-1900, 145-179. The line of iV' = 532(p. 17) is incorrect. 
"Math. Quest. Educat. Times, 73, 1900, 83-94. [Some errors.] 
"Mem. Pont. Ac. Nuovi Lincei, 17, 1900, 321-344; errata, 18, 1901. Cf. Sphinx-Oedipe, 5, 
1910, num6ro special, 1-9. Cf. Jahrbuch Fortschritte Math., on a = 37. 
*»Atti Accad. Pont. Nuovi Lincei, 44, 1900-1, 89. 
