418 History of the Theory of Numbers. [Chap, xviii 
H. Weber^^ and E. Schering^^ completed Dirichlet's^^ proof of his first 
theorem. A. Meyer^^ completed Dirichlet's^^ proof of his extended theorem. 
F. Mertens^" gave an elementary proof of Dirichlet's^^ extended theorem. 
Ch. de la Vall^e-Poussin^^ proved that the number of primes ^ x repre- 
sentable by a properly primitive definite positive or indefinite^^ irreducible 
binary' quadratic form is asjTnptotic to gx/\ogx, where ^ is a constant; and 
the same for primes belonging also to a linear form compatible with the 
character of the quadratic form. 
L. Kronecker^ (pp. 494-5) stated a theorem on factorable forms in 
several variables which represent an infinitude of primes. 
Elementary Proofs of the Existence of an Infinitude of Primes mz+l, 
FOR Any Given m. 
V. A. Lebesgue^^ gave a proof for the case m a prime, using the fact 
that x^~^ — x"'~~y+ . . . +y"'~^ has besides the possible factor m only prime 
factors 2km + 1. A like method apphes^^ to 2mz — l. 
J. A. Serret^^ gave an incomplete proof for any m. 
F. Landry^^ gave a proof like Lebesgue's.®^ If ^ is the largest prime 
2km-\-l and if x is the product of all of them, a:'"+l is divisible by no one 
of them. Since (x'"+l)/(x+l) has no prime divisor not of the form 
2km+l, there exists at least one >d. 
A. Genocchi^^ proved the existence of an infinitude of primes mz^ 1 
and n'2±l for n a prime by use of the rational and irrational parts of 
(a+Vb)'- 
L. Kronecker^ (pp. 440-2) gave in lectures, 1875-6, a proof for the case 
m a prime; the simple extension in the text to any m was added by Hensel. 
E. Lucas gave a proof by use of his w„ (Lucas,^^ p. 291, of Ch. XYII). 
A. Lefebure^° of Ch. XVI stated that the theorem follows from his 
results. 
L. Kraus^^ gave a proof. 
A. S. Bang"'' and Sylvester^ proved it by use of cyclotomic functions. 
K. Zsigmondy"^ of Ch. VII gave a proof. Also, E. Wendt,'^ and 
Birkhoff and Vandiver^^ ^f q^ ^VI. 
»^Math. Annalen, 20, 1882, 301-329. Elliptische Functionen (= Algebra, III), ed. 2, 1908, 
613-6. 
»8Werke, 2, 1909, 357-365, 431-2. 
"Jour, fur Math., 103, 1888, 98-117. Exposition by Bachmaim,»< pp. 272-307. 
•oSitzungsber. Ak. Wiss. Wien (Math.), 104, 1895, Ila, 1093-1153, 1158. Simplification, 
ibid., 109, 1900, Ila, 415-480. 
"Cf. E. Landau, Jahresber. D. Math. Verein., 24, 1915, 250-278. 
"Jour, de Math., 8, 1843, 51, note. Exercices d'analyse num^rique, 1859, 91. 
«''.Jour. de Math., (2), 7, 1862, 417. 
•«Jour. de Math., 17, 1852, 186-9. 
•'Deuxidme mdmoire but la thdorie des nombres, Paris, 1853, 3. 
"AnnaU di mat., (2), 2, 1868-9, 256-7. Cf. .Genocchi==- " of Ch. XVII. 
••Casopis Math, a Fys., 15, 1886, 61-2. Cf. Fortschritte, 1886, 134-5. 
^Tidsskrift for Math., (5), 4, 1886, 70-80, 130-7. See Bang". ", Ch. XVI. 
"Jour, fiir Math., 115, 1895, 85. 
i 
