386 HISTORY OP THE THEORY OF NUMBERS. ICSAP. xvi E. Lucas43 gave algebraic factors of x12+6V2. K. Zsigmondy44 proved the existence of a prime dividing (P—fr, but no similar binomial with a lower exponent, exceptions apart (cf . Bang,33* 34 Birkhoff62). J. W. L. Glaisher45 gave the prime factors of p6--(-l)(p-1)/2 for each prime p<100. T. Pepin46 proved that (317-1)/30, (835-l)/82, (241+l)/(3-83) are primes. A. A. Markoff47 investigated the greatest prune factor of n2+l. W. P. Workman48 noted the factors of 36fc+3+l [due to Catalan28] and 254+l, and stated that Lucas43 (p. 326) gave erroneous factors of 258+l. C. E. Bickmore49 gave factors of an-l for ng50, a =2, 3, 5, 6, 7, 10, 11, 12. Several490 proved that nn — 1 is divisible by 4n+l if 4r&+l is prime. A. Cunningham50 gave 43 primes exceeding 9 million which are factors of (s5=fcl)/(a;=bl), and factors of 330+1, 333-l, 3*+l, 3105+1, 513-1, 514+1, 517-1, 520+1, 535~1. A. Cunningham51 considered at length the factorization of Aurif euillians, t. e.t the algebraically irreducible factors of where % and x are relatively prime to n% and y, while n has no square factor, .and is odd in the second case. Aurifeuille had found them to be expressible algebraically in the form P2— Q2. There are given factors of 2n+l for n even and ^102, and for n=110, 114, 126, 130, 138, 150, 210. A. Cunningham52 factored numbers an=*= 1 by use of tables, complete to y « 101, giving the lengths I of the periods of primes p and their powers < 10000 to various bases g, so that g*s=l (mod p or pk). A. Cunningham and H. J. Woodall53 gave factors of 2V=2x10a=*=l for s^s30, aglO, and for further sets; also, for each prime p ^5 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 N. T. Pepin64 factored a7-! for a=37, 41, 79; also55 1516-1. ^TlwSorie des nombres, 1891, 132, exs. 2-4. "Monatshefte Math. Phys., 3, 1892, 283. Details in Ch. VII, Zsigmondy.78 «Quar. Jour. Math., 26, 1893, 47. "Memorie Accad. Pont. Nuovi Lincei, 9, 1, 1893, 47-76. <7Comptes Rendus Paris, 120, 1895, 1032. ^Messenger Math., 24, 1895, 67. «/Kd., 25, 1896, 1-44; 26, 1897, 1-38; French transl., Sphinx-Oedipe, 7, 1912, 129-44, 155-9. «aMath. Quest. Educ. Times, 65, 1896, 78; (2), 8, 1905, 97. «0Proc. London Math. Soc., 28, 1897, 377, 379. "Ibid., 29, 1898, 381-438. "Messenger Math., 29, 1899-1900, 145-179. The line of AT' = 532(p. 17) is incorrect. B3Math. Quest. Educat, Times, 73, 1900, 83-94. [Some errors.] MMem. Pont. Ac. Nuovi Lincei, 17, 1900, 321-344; errata, 18, 1901. Cf. Sphinx-Oedipe, 5, 1910, nume'ro special, 1-9. Cf. Jahrbuch Fortschritte Math., on a=«37. 66Atti Accad. Pont. Nuovi Lincei, 44, 1900-1, 89.