382 History of the Theory of Numbers. [Chap, xvi 
Euler^" discussed the divisors of numbers of the iorm.fa^+gb'^. 
Anton FelkeP" gave a table, incomplete as to a few entries, of the factors 
ofa"-l,n = l,..., ll;a = 2,3,..., 12. 
A. M. Legendre^^ proved that every prime divisor of a"+l is either of the 
form 2nx-{-l or divides a"H-l where co is the quotient of n by an odd factor; 
every prime divisor of a" — ! is either of the form nx-\-l or divides a'^ — l 
where co is a factor of n. For n odd, the divisors must occur in 0(0"=^ 1) 
= ?/'-± a and are thus further limited by his tables III-XI of the linear forms 
of the divisors of <"±au". 
C. F. Gauss^^ obtained by use of the quadratic reciprocity law the linear 
forms of the divisors of x"— A. 
Gauss^'^ gave a table of 2452 numbers of the forms a^+1, a^+4, . . ., 
a^+81 and their odd prime factors p, for certain a's for which the p's are 
all <200. 
Sophie Germain^^ noted that p'*+4g* has the factors p^=^2pq+2^ 
[Euler^]. Taking p = l, q = 2\ we see that 2^'"''^+l has the two factors 
22.+i±2'+i + l. 
F. Minding^^ gave a detailed discussion of the linear forms of the divisors 
of x^ — c, using the reciprocity law for the case of primes. He reproduced 
(pp. 188-190) the discussion by Legendre.^^ 
P. L. Tchebychef^® noted that, if p is an odd prime, every odd prime 
factor of a''— 1 is either of the form 2pz-\-l or is a factor of a — 1, and more- 
over is a divisor of x^ — ay'^. Hence, for a = 2, it is of the form 2^2+1 and 
also of one of the forms 8w=tl. Every odd prime factor of a^^+^ + l is 
either of the form 2(2n+ 1)2+1 or a divisor of a+1 [cf. Legendre"]. 
V. A. Lebesgue^^ noted that the discussion of the linear forms of the 
divisors of z^—D, where D is composite, is simplified by use of Jacobi's 
generalization (a/b) of Legendre's symbol. 
C. G. Reuschle^' denoted (x"''-l)/(x''-l) by FM. Set a = ah+bi, 
6 = 0161+62, 61 = 0262+63,. ... If a, 6 are relatively prime, 
(T^ ^r)C n = '^'^""nia(6-l-A)l+a;^ sVxjai(6i-l-A)| 
{X —i){X —L) A=0 A=0 
+ ... +a:^+^.+ - ■ -+^-2 Vx^'"-'n„_i {a._i(6„_i -1-A)} +x^+- • ■+^n-i. 
9^'Opera postuma, I, 1862, 161-7 (about 1773). 
loAbhandl. d. Bohmischen Gesell. Wiss., Prag, 1, 1785, 165-170. 
'•Th6orie des nombres, 1798, pp. 207-213, 313-5; ed. 2, 1808, pp. 191-7, 286-8. German 
transl. by Mascr, p. 222. 
"Disq. Arith., 1801, Arts. 147-150. 
"Werke, 2, 1863, 477-495. Schering, pp. 499-502, described the table and its formation by 
the compo.sition of binary forms, e. g., (a^' + l) { (a4-l)^4-l} = {a(o + l)+l}* + l. 
"Manuscript 9118 fonds frangais Bibl. Nat. Paris, p. 84. Cf. C. Henry, Assoc, frang. avanc. 
sc, 1880, 205; Oeuvres de Fermat, 4, 1912, 208. 
"Anfangsgrunde dor Hoheren Arith., 1832, 59-70. 
"Theorie der Congrucnzen, in Russian, 1849; in German, 1889; §49. 
"Jour, de Math., 15, 1850, 222-7. 
"Math. Abhandlung, Stuttgart, 1853, II, pp. 6-13. 
