CHAP, xvi] FACTORS OF an^b\ 383 Reuschle's19 table A gives many factors of a3=±=l, a4dt=l, a5=*=l, a12— 1 for a^lOO, and of att-l for n^42, a = 2, 3, 5, 6, 7, 10. Lebesgue19* proved that xp~l+ . . .+#+1 has no prime divisor other than the prime p and numbers of the form kp+1. Jean Plana20 gave 329+l=4-609l2, 329-l = 2-59r, and stated that q is a prune and that r has no factor < 52259. But Lucas26 noted that q = 523-5385997, r = 28537-20381027. E. Kununer21 proved that there is no prime factor, other than t and numbers 2m£=fcl, of the cyclotomic function obtained from (a'—l)/(a — 1) by setting a 4-fl~"1:=£, t being a prime 2e+l-E. Catalan22 stated that, if n = a=Fl is odd, an=pl is divisible by n2, but not by n3. Proof by Soons, Mathesis, (3), 2, 1902, 109. H. LeLasseur and A. Aurifeuille23 noted that 24n+2-fl has the factors 22n+i=fc2n+i+1 [cf> Eulerj2 Sa Germain14]. E. Lucas24 proved that (240+l)/(28-}-l) is a prime and gave the factors of Theorems by Lucas on the factors of an=*=6n, given in various papers in 1876-8, are cited in Ch. XVII. Lucas25 factored (2m)m±l for m = 7, 10, 11, 12, 14, 15, and corrected Plana.20 Lucas26 gave tables due to LeLasseur and Aurifeuille of functions (nodd)f ^ " expressed in the form Y^^pxyZ2, which is factorable if xy = ptf*. Factors of xw+yl° are given for various x's, y's. He gave LeLasseur's table of the proper divisors of 2n— 1 for all odd values of n<100 except n = 61, 67, 71, 77, 79, 83, 85, 89, 93, 97; the proper divisors of 2n+l for n odd and <71 (except ft = 61, 67) and for n = 73, 75, 81, 83, 99, 135; the proper divisors of 22*+l for 2/c^74 (except 64, 68) and for 2fc = 78, 82, 84, 86, 90, 94, 102, 126, etc. Lucas proved (pp. 790-4) that the proper divisors of 24n+l are of the form 16ng+l, those of a2abn+b2abn are of the form 8abnq+l; for n odd, those of aabn+babn are of the form 4abng+l if a6 = 4/i+l, those of aabn-babn are of the form 4afcn#+l if ab = 4h+3. "Math. Abhandlung.. . .Tabellen, Stuttgart, 1856. Full title in Ch. I. 19aComptes Rendus Paris, 51, 1860, 11. MMem. Accad. Sc. Torino, (2), 20, 1863, 139-141. 21Cf. Bachmann, Kreistheilung, Leipzig, 1872. 22Revue de Tlnstruct. publique en Belgique, 17, 1870, 137; Melanges Math., ed. 1, p. 40. "Atti R. Ac. Sc. Torino, 8, 1871; 13, 1877-8, 279. Nouv. Corresp. Math., 4, 1878, 86, 98. Cf. Lucas,26 p. 238; Lucas,28 784. MNouv. Ann. Math.', (2), 14, 1875, 523-5. 26Amer. Jour. Math., 1, 1878, 293. ^Bull. Bibl. Storia Sc. Mat. e Fis., 11, 1878, 783-798.