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Full text of "History Of The Theory Of Numbers - I"

CHAP, xviii]                      INFINITUDE OF PRIMES.                                 415
G. Me*trod17 noted that the sum of the products n1 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.