388 History of the Theory of Numbers. [Chap, xvi 
P. F. Teilhet*^ gave formulas factoring cases of l+ar^, as 
(6H6+l)Hl = [(6+l)2-hl](6Hl), 
4(c+ir+l =[(c+2)H(c+l)^[(c+l)Hca 
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). 
Escott^° repeated Euler's^ remarks on the integers x for which \-\-:i^ 
is di\'isible by a given prime. He and Teilhet (11, 1904, 10, 203) noted 
that any common di\'isor of h and a±l di\'ides (a''='=l)/(a=fcl). 
G. Wertheim" collected the theorems on the divisors of a"'=«=l. 
G. D. Birkhoff and H. S. Vandiver^- employed relatively prime integers 
a, 6 (a>6) and defined a primitive divisor of F„ = a" — 6" to be one relatively 
prime to V^, for all di\'isors m of n. They proved that, if n?^2, F„ has a 
primitive divisor 7^ 1 except f orn = 6, a = 2, 6 = 1. 
L. E. Dickson^^'' noted that {p^ — l){p^ — l) has no factor=l (mod p^) 
if p is prime. 
A. Cunningham^^ gave high primes ?/" + l, (t/^+1)/2, ?/^4-y+l. 
H. J. Woodall^ gave factors of ?/" + l. 
J. W. L. Glaisher^^ factored 2^''=t2^+l for r^ll, in connection with 
the question of the similarity of the nth pedal triangle to a given triangle. 
L. E. Dickson^^ gave a new derivation of (1), found when F,(a) is divisible 
by pi or pi^, where pi is a prime factor of t, and proved that, if a is an integer 
>1, F,(a) has a prime factor not di\'iding 0*^ — 1 (m<t) except in the cases 
f = 2, a = 2* — 1, and t = Q, a = 2; whence a' — 1 has a prime factor not dividing 
a"*—l{m<t) except in those cases [cf. Birkhoff,^- CarmichaeP]. 
Dickson®^ applied the last theorem to the theory of finite algebras and 
gave material on the factors of p" — 1. 
A. Cunningham^^ treated at length the factorization of i/"+l for 71 = 2, 
4, 8, 16, and (?/^''+l)/(2/"+l) for n = 1, 2, 4, 8, by means of extensive tables 
of solutions of the corresponding congruences modulo p. He discussed also 
x^+y", n = 4, 6, 8, 12. 
Cunningham^^'' factored \{x^ — i^)/{x — y)-\-iJL{x^+y^)/{x'^+y'^) by ex- 
pressing the fractions in the form P^ — kxyQ"^, k= o, 6. 
"L'intermediaire des math., 9, 1902, 31&-8. 
^oibid., 12, 1905, 38; cf. 11, 1904, 195-6. 
"Anfangsgriinde der Zahlenlehre, 1902, 297-303, 314. 
"Annals of Math., 5, 1903-4, 173. Cf. Zsigmondy," Dickson." 
•«»Amer. Math. Monthly, 11, 1904, 197, 238; 15, 1908, 90-1. 
•»Quar. Jour. Math., 35, 1904, 10-21. 
•*Ibid., p. 95. 
**Ibid., 36, 1905, 156. 
"Amer. Math. Monthly, 12, 1905, 86-89. 
•'Gottingen Nachrichten, 1905, 17-23. 
"Messenger Math., 35, 1905-6, 16&-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^; (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. 
•»<»Math. Quest. Educ. Times, 10, 1906, 58-9. 
I 
