Chap. XVIII] INFINITUDE OF PrIMES. 415 
G. M^trod^'^ noted that the sum of the products n — 1 at a time of the 
first n prunes > 1 is either a prime or is divisible by a prime greater than the 
nth. He also repeated Euler's^ proof. 
Infinitude of Primes in a General Arithmetical Progression. 
L. Euler^'' stated that an arithmetical progression with the first term 
unity contains an infinitude of primes. 
A. M. Legendre^^ claimed a proof that there is an infinitude of primes 
2mx+ii if 2m and /x are relatively prime. 
Legendre^^ noted that the theorem would follow from the following 
lemma: Given any two relatively prime integers A, C, and any set of k odd 
primes d,\,. . ., co [not divisors of A], and denoting the zth. odd prime by 
7r^^\ then among tt'*"^' consecutive terms of the progression A — C, 2A — C, 
3A — C,. . . there occurs at least one divisible by no one of the primes 
6,. . ., CO. Although Legendre supposed he had proved this lemma, it is 
false [Dupr^28j^ 
G. L. Dirichlet^^ gave the first proof that mz-\-n represents infinitely 
many primes if m and n are relatively prime. The difficult point in the 
proof is the fact that 
n=i n 
where x(^) =0 if ^j ^ have a common factor > 1, while, in the contrary case, 
xin) is a real character different from the chief character of the group of 
the classes of residues prime to k modulo k. This point Dirichlet proved 
by use of the classes of binary quadratic forms. 
Dirichlet^'* extended the theorem to complex integers. 
E. Heine^^ proved "without higher calculus" Dirichlet's result 
VJ? p[jHW^''^{b+2ay+'>~^ ■■■}'= a 
A. Desboves^^ discussed the error in Legendre's^^ proof. 
L. Durand^^ gave a false proof. 
A. Dupre^^ showed that the lemma of Legendre^" is false and gave 
(p. 61) the following theorem to replace it: The mean number of terms, 
"L'intermediaire des math., 24, 1917, 39-40. 
^oOpusc. analytica, 2, 1785 (1775), 241; Comm. Arith., 2, 116-126. 
2iMem. ac. sc. Paris, ann^e 1785, 1788, 552. 
22Theorie des nombres, ed. 2, 1808, p. 404; ed. 3, 1830, II, p. 76; Maser, 2, p. 77. 
23Bericht Ak. Wiss. Berlin, 1837, 108-110; Abhand. Ak. Wiss. Berlin, Jahrgang 1837, 1839, 
Math., 45-71; Werke, 1, 1889, 307-12, 313-42. French transl., Jour, de Math., 4, 1839, 
393-422. Jour, fiir Math., 19, 1839, 368-9; Werke, 1, 460-1. Zahlentheorie, §132, 1863; 
ed. 2, 1871; 3, 1879; 4, 1894 (p. 625, for a simplification by Dedekind). 
z^Abhand. Ak. Wiss. BerUn, Jahrgang 1841, 1843, Math., 141-161; Werke, 1, 509-532. French 
transl., Jour, de Math., 9, 1844, 245-269. 
2«Jour. fiir Math., 31, 1846, 133-5. 
26Nouv. Ann. Math., 14, 1855, 281. 
^Ubid., 1856, 296. 
^^Examen d'une proposition de Legendre, Paris, 1859. Comptes Rendus Paris, 48, 1859, 487. 
