History Of The Theory Of Numbers - I

CHAP, xviii] INFINITUDE OF PRIMES. 415
G. Me*trod17 noted that the sum of the products n—1 at a time of the first n primes > 1 is either a prime or is divisible by a prime greater than the nth. He also repeated Euler's4 proof.
INFINITUDE OF PKIMES IN A GENERAL ARITHMETICAL PROGRESSION.
L. Euler20 stated that an arithmetical progression with the first term unity contains an infinitude of primes.
A. M. Legendre21 claimed a proof that there is an infinitude of primes 2mx+n if 2m and ju are relatively prime.
Legendre22 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 6, X,..., co [not divisors of A], and denoting the zth odd prune by T('\ then among T^""1' consecutive terms of the progression A —C, 2A — C, 3A — C,... there occurs at least one divisible by no one of the prunes 6,..., co. Although Legendre supposed he had proved this lemma, it is false [Dupre*28].
G. L. Dirichlet23 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
where x(ri) = 0 if n, k have a x(n) is a real character diff the classes of residues prime — by use of the classes of binary q\ Dirichlet24 extended the theoren E. Heine26 proved "without
A. Desboves26 discussed the error hi Legendre's22 proof. L. Durand27 gave a false proof.
A. Dupre"28 showed that the lemma of Legendre22 is false and gave (p. 61) the following theorem to replace it: The mean number of terms,
"L'intenruSdiaire des math., 24, 1917, 39-40.
»°0pusc, analytica, 2, 1785 (1775), 241; Comm. Arith., 2, 116-126.
21Me*m. ac. so. Paris, anne*e 1785, 1788, 552.
^ThSorie des nombres, ed. 2, 1808, p. 404; ed. 3, 1830, II, p. 76; Maser, 2, p. 77.
"Bericht 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). 2<Abhand. Ak. Wiss. Berlin, Jahrgang 1841, 1843, Math., 141-161; Werke, 1, 509-532. French
transl., Jour, de Math., 9, 1844, 245-269. 26 Jour, fiir Math., 31, 1846, 133-5. 2eNouv. Ann. Math., 14, 1855, 281. "Ibid., 1856, 296. 28Examen d'une proposition de Legendre, Paris, 1859. Comptes Rendus Paris, 48, 1859, 487.