Chap. XVI] FACTORS OF a" ±6". 383 
Reuschle's^^ table A gives many factors of a^=tl, a*±l, a^±l, a^^ — 1 
for a^lOO, and of a"-l for n^42, a = 2, 3, 5, 6, 7, 10. 
Lebesgue^^" proved that x*^^+ . . .-\-x-\-l has no prime divisor other 
than the prime p and numbers of the form kp + 1. 
Jean Plana^^ gave S^^ + l =4-6091g, S^^-l=2-59r, and stated that q 
is a prime and that r has no factor < 52259. But Lucas^^ noted that 
q = 523-5385997, r = 28537-20381027. 
E. Kummer^^ proved that there is no prime factor, other than t and 
numbers 2m^±l, of the cyclotomic function 
obtained from (a' — l)/(a — 1) by settinga+a~^ = a:;, t being a prime 2e + l. 
E. Catalan^^ stated that, if n = a=Fl is odd, a"=Fl is divisible by n^, but 
not by nK Proof by Soons, Mathesis, (3), 2, 1902, 109. 
H. LeLasseur and A. Aurifeuille^^ noted that 2^"'''^ + l has the factors 
22n+i± 2^^+1 + 1 [cf. Euler,2 S. Germain^^]. 
E. Lucas^^ proved that (2^°+l)/(2^+l) is a prime and gave the factors of 
30^^±1, 2*^ + 1. 
Theorems by Lucas on the factors of a" ±6**, given in various papers in 
1876-8, are cited in Ch. XVII. 
Lucas^^ factored (2m)"'=tl for m = 7, 10, 11, 12, 14, 15, and corrected 
Plana.20 
Lucas^^ gave tables due to LeLasseur and Aurifeuille of functions 
5!il! („odd), ^^, 
x^y x^-i-y^ 
expressed in the form V^^pxyZ^, which is factorable if xy = ptP. Factors 
of x^°-\-y^^ are given for various x's, y's. He gave LeLasseur's table of the 
proper divisors of 2" — 1 for all odd values of n<100 except n = 61, 67, 
71, 77, 79, 83, 85, 89, 93, 97; the proper divisors of 2"+l for n odd and <71 
(except n = 61, 67) and for n = 73, 75, 81, 83, 99, 135; the proper divisors 
of 22^1 for 2A;^74 (except 64, 68) and for 2k = 7S, 82, 84, 86, 90, 94, 102, 
126, etc. Lucas proved (pp. 790-4) that the proper divisors of 2^"+l are of 
the form 16ng + l, those of (j2a6n_|_^2a67i ^^.^ q£ ^^^q form Sahnq-j-l; for n odd, 
those of a"^"+6"^'^ are of the form 4a6ng + l if a6 = 4/i + l, those of a"''" -6"'''* 
are of the form 4a6ng + l if a6 = 4/i+3. 
I'Math. Abhandlung . . .Tabellen, Stuttgart, 1856. Full title in Ch. I. 
"aComptes Rendus Paris, 51, 1860, 11. 
20Mem. Accad. Sc. Torino, (2), 20, 1863, 139-141. 
"Cf. Bachmann, Kreistheilung, Leipzig, 1872. 
22Revue de I'lnstruct. publique en Belgique, 17, 1870, 137; Melanges Math., ed. 1, p. 40. 
23Atti R. Ac. Sc. Torino, 8, 1871; 13, 1877-8, 279. Nouv. Corresp. Math., 4, 1878, 86, 98. 
Cf. Lucas,25 p. 238; Lucas,^^ 784. 
2^Nouv. Ann. Math., (2), 14, 1875, 523-5. 
25Amer. Jour. Math., 1, 1878, 293. 
26Bull. Bibl. Storia Sc. Mat. e Fis., 11, 1878, 783-798. 
