388 HlSTOBY OP THE THEORY OF NUMBERS. [CHAP. XVI P. F. Teilhet69 gave formulas factoring cases of 1+x2, as the last being (10, 1903, 170) a case of the known formula for the product of two sums of two squares (cf. 11, 1904, 50). Escott60 repeated Euler's7 remarks on the integers x for which 1+s2 is divisible by a given prime. He and Teilhet (11, 1904, 10, 203) noted that any common divisor of 6 and a=*=l divides (a&=±=l)/(a=t=l). G. Wertheim61 collected the theorems on the divisors, of am=*=l. G. D. Birkhoff and H. S. Vandiver62 employed relatively prime integers a, 6 (a>b) and defined a primitive divisor of Vn=an— bn to be one relatively prime to Vmj for all divisors m of ft. They proved that, if n?^2, Vn has a primitive divisor 5^1 except for n = 6, a = 2, 6 = 1. L. E. Dickson620 noted that (p4-l)(p2-!) has no factor=l (mod p3) if p is prime. A. Cunningham63 gave high primes y2+l, (y*+l)/2, y*+y+l. H. J. Woodall64 gave factors of y*+l. J. W. L. Glaisher65 factored 22r=*=2r-H for r^ll, in connection with the question of the similarity of the nth pedal triangle to a given triangle. L. E. Dickson66 gave a new derivation of (1), found when Ft(a) is divisible by PI or pi2, where PI is a prime factor of t, and proved that, if a is an integer >1, Ft(a) has a prime factor not dividing am— 1 (m<t) except in the cases t = 2, a = 2k — 1, and t = 6, a = 2 ; whence a' — 1 has a prime factor not dividing am— 1 (m<t) except in those cases [cf. Birkhoff,62 Carmichael78]. Dickson67 applied the last theorem to the theory of finite algebras and gave material on the factors of pn— 1. A. Cunningham68 treated at length the factorization of ?/n+l for n = 2, 4, 8, 16, and (y*n+l)/(yn+l) for n = l, 2, 4, 8, by means of extensive tables of solutions of the corresponding congruences modulo p. He discussed also xn+yn, n = 4, 6, 8, 12. Cunningham68a factored \(x*-y*)/(x-y)+ijl(x«+y6)/(x*+y2) by expressing the fractions in the form P2~kxyQ*, k=5, 6. 69I/interm£diaire dea math., 9, 1902, 316-8. «°Ibid., 12, 1905, 38; cf. 11, 1904, 195-6. fllAnfangsgriinde der Zahlenlehre, 1902, 297-303, 314. "Annals of Math., 5, 1903-4, 173. Cf. Zsigmondy," Dickson.64 a^Amer. Math. Monthly, 11, 1904, 197, 238; 15, 1908, 90-1. MQuar. Jour. Math., 35, 1904, 10-21. "/bid., p. 95. «/Wd.f 36, 1905, 156. •"Arner. Math. Monthly, 12, 1905, 86-89. «7G6ttingen Nachrichten, 1905, 17-23. "Messenger Math., 35, 1905-6, 166-185; 36, 1907, 145-174; 38, 1908-9, 81-104, 145-175; 39, 1909, 33-63, 97-128; 40, 1910-11, 1-36. Educat. Times, 60, 1907, 544; Math. Quest. Educat. Times, (2), 13, 1908, 95-98; (2), 14, 1908, 37-8, 52-3, 73-4; (2), 15, 1909, 33-4, 103-4; (2), 17, 1910, 88, 99. Proc. London Math. Soc., 27, 1896, 98-111; (2), 9, 1910, 1-14. h. Quest. Educ. Times, 10, 1906, 58-9.