History Of The Theory Of Numbers - I

CHAP, xviii] , INFINITUDE OF PRIMES. 419
N. V. Bervi72 proved that the ratio of the number of integers cm-f-1 not >n and not a product of two integers of that form to the number of all primes not >n has the limit unity for 71= «>.
H. C. Pocklington73 proved that, if n is any integer, there is an infinitude of primes mn+l, an infinitude not of this form if ft > 2, and an infinitude not of the forms mn^=l if n=5 or n>6.
E. Cahen74 proved the theorem for m an odd prime.
J. G. van der Corput75 proved the theorem.
ELEMENTARY PROOFS OF THE EXISTENCE OF AN INFINITUDE OF PRIMES IN SPECIAL ARITHMETICAL PROGRESSIONS.
J. A. Serret6Mor the common difference 8 or 12, and for lOz+9.
V. A. Lebesgue80 for 4ft=*=l, 8n+k (fc = l, 3, 5, 7). Lebesgue8* for the same and 2wn.+l, 6n—1. Also, by use of infinite series, for the common difference 8 or 12.
E. Lucas82 for 5n+2, Sn+7.
J. J. Sylvester83 for the difference 8 or 12 arid84 for pkx—1, p a prime.
A. S. Bang85 for the differences 4, 6, 8, 10, 12, 14, 18, 20, 24, 30, 42, 60.
E. Lucas86 for 4n±l, 6n-l, 8n+5.
R. D. von Sterneck87 for an—I.
K Th. Vahlen88 for mz+l by use of Gauss' periods of roots of unity. Also, if m is any integer and p a prime such that r> — 1 is divisible bv a hiVhAr power of 2 than <i>(rn) is, while form mpx+km+l represents ai are mx+l and 2px—1.
J. J. Iwanow89 for the difference 8 t
E. Cahen14 (pp. 318-9) for 4z=*=l, 6x^ _, _ , _. 508) for the same forms. M. Bauer90 for an — I.
E. Landau47 (pp. 436-46) for kn^l.
I. Schur91 proved that if Z2s= 1 (mod k) and if one knows a prime ><£(fc)/2 of the form kz+l, there exists an infinitude of primes kz+l; for example,
2nz+2ft~1=±=l, Smz+2m+l, 8mz+4w+l, 8m2+6m+l,
where m is any odd number not divisible by a square.
K. Hensel92 for 4n=*=l, 6n±l, 8n-l, 8n=«=3, 12n-l, 10n-l.
72Mat. Sbornik (Math. Soc. Moscow), 18, 1896, 519.
7«Proc. Cambr. Phil. Soc., 16, 1911, 9-10. 74Nouv. Ann. Math., (4), 11,1911, 70-2.
76Nieuw Archief voor Wiskunde, (2), 10, 1913, 357-361 (Dutch).
80Nouv. Ann. Math., 15, 1856, 130, 236.
81Exercicesd'analysenum6rique, 1859,91-5,103-4,145-6.
82Amer. Jour. Math., 1,1878,309. ^Comptes Rendus Paris, 106, 1888, 1278-81, 1385-6.
"Assoc. franQ. av. sc., 17, 1888, II, 118-120.
MNyt Tidsskrift for Math., Kjobenhavn, 1891, 2B, 73-82.
MTh6orie des nombres, 1891, 353nt. "Monatahefte Math. Phys., 7, 1896, 46,
88Schriften phys.-okon. Gesell. Konigsberg, 38, 1897, 47.
89Math. Soc. St. Petersburg, 1899, 53-8 (Russian).
BOJour. fur Math., 131, 1906, 265-7; transl. of Math. e*s Phys. Lapok, 14, 1905, 313.
"Sitzungsber. Berlin Math. Gesell., 11, 1912, 40-50, with Archiv M. P.
wZahlentheorie, 1913, 304-5.