382 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvi
Euler9* discussed the divisors of numbers of the form/a4+#b4.
Anton Felkel10 gave a table, incomplete as to a few entries, of the factors of an-l, n=l, . . ., 11; a=2, 3,. . ., 12.
A. M. Legendre11 proved that every prime divisor of an+l is either of the form 2nx+l or divides aw+l where co is the quotient of n by an odd factor; every prime divisor of an— 1 is either of the form nx+l or divides aw— 1 where co is a factor of n. For n odd, the divisors must occur in a(an=*= 1) = 2/2=t= a and are thus further limited by his tables III-XI of the linear forms of the divisors of f^au2.
C. F. Gauss12 obtained by use of the quadratic reciprocity law the linear forms of the divisors of x2—A.
Gauss13 gave a table of 2452 numbers of the forms a2+l, a2 +4, . . ., a2 +81 and their odd prime factors p, for certain a's for which the p's are all <200.
Sophie Germain14 noted that p4+4g4 has the factors p2*=2pq+2q2 [Euler2]. Taking p = l, g=2\ we see that 24i+2+l has the two factors 22t"+1=fc2t"+1+l.
F. Minding15 gave a detailed discussion of the linear forms of the divisors of x2— ct using the reciprocity law for the case of primes. He reproduced (pp. 188-190) the discussion by Legendre.11
P. L. Tchebychef16 noted that, if p is an odd prime, every odd prime factor of ap— 1 is either of the form 2pz+l or is a factor of a— 1, and moreover is a divisor of x2—ay2. Hence, for a = 2, it is of the form 2pz+l and also of one of the forms 8ra=t=l. Every odd prime factor of a2n+1+l is either of the form 2(2n+ 1)2+1 or a divisor of a+1 [cf. Legendre11].
V. A. Lebesgue17 noted that the discussion of the linear forms of the divisors of z2— D, where D is composite, is simplified by use of Jacobi's generalization (a/6) of Legendre' s symbol.
C. G. Reuschle18 denoted (zab-l)/(z°-l) by Ftt(6). Set a = a&+&lf 6 = a1&1+&2, ^1 = ^262 +63,. ... If a, 6 are relatively prime,
— ^=|^
flaOpera postuma, I, 1862, 161-7 (about 1773).
10Abhandl. d. Bohniiachen Gesell. Wiss., Prag, 1, 1785, 165-170.
"Th^orie des nombres, 1798, pp. 207-213, 313-5; ed. 2, 1808, pp. 191-7, 286-8. German
transl. by Maser, p. 222. 12Disq. Arith., 1801, Arts. 147-150. "Werke, 2, 1863, 477-495. Sobering, pp. 499-502, described the table and its formation by
the composition of binary forms, e. g., (a2-fl){(a-fl)z-f-l} = (a(a+l)-f l}2-fl. "Manuscript 9118 fonds francais Bibl. Nat. Paris, p. 84. Cf. C. Henry, Assoc. franc,, avanc.
sc., 1880, 205; Oeuvres de Fermat, 4, 1912, 208. "Anfangsgriinde der Hoheren Arith., 1832, 59-70. "Theorie der Congruenzen, in Russian, 1849; in German, 1889; §49. "Jour, de Math., 15, 1850, 222-7. "Math. Abhandlung, Stuttgart, 1853, II, pp. 6-13.