Chap. XVIII] INFINITUDE OF PRIMES. 419 
N. V. Bervi^^ proved that the ratio of the number of integers cm+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 n= oo . 
H. C. Pocklington^^ proved that, if n is any integer, there is an infinitude 
of primes mn+1, an infinitude not of this form if n>2, and an infinitude 
not of the forms mn±l if n = 5 or n>6. 
E. Cahen^^ proved the theorem for m an odd prime. 
J. G. van der Corput'^^ proved the theorem. 
Elementary Proofs op the Existence of an Infinitude of Primes in 
Special Arithmetical Progressions. 
J. A. Serret^^ for the common difference 8 or 12, and for lOx+9. 
V. A. Lebesgue»° for 4n±l, Sn+k (/c = l, 3, 5, 7). Lebesgue^^ for the 
same and 2'"n+l, Qn — 1. Also, by use of infinite series, for the common 
difference 8 or 12. 
E. Lucas«2 f<3j. 5n-\-2, 8n+7. 
J. J. Sylvester^^ for the difference 8 or 12 and^ for p^x — 1, p a prime. 
A. S. Bang^^ for the differences 4, 6, 8, 10, 12, 14, 18, 20, 24, 30, 42, 60. 
E. Lucas^^ for 4n=i=l, 6n — 1, Sn+5. 
R. D. von Sterneck" for an — 1. 
K. Th. Vahlen^^ for mz-\-l by use of Gauss' periods of roots of unity. 
Also, if m is any integer and p a prime such that p — 1 is divisible by a higher 
power of 2 than <^(m) is, while A; is a root of km + 1^ —1 (mod p), the linear 
form mpx+km-^-l represents an infinitude of primes; known special cases 
are mx+1 and 2px — l. 
J. J. Iwanow^^ for the difference 8 or 12. 
E. Cahen^^ (pp. 318-9) for 4a;± 1, 6x± 1, 8a:+5. K. HenseP (pp. 438-9, 
508) for the same forms. M. Bauer^° for an — 1. 
E. Landau^^ (pp. 436-46) for /cn±l. 
I. Schur^^ proved that if f= 1 (mod k) and if one knows a prime '>(f>(k)/2 
of the form kz+l, there exists an infinitude of primes kz+l; for example, 
2»z-\-2''-^=i=l, Smz-\-2m+l, Smz+4m-\-l, 8m2+6m+l, 
where m is any odd number not divisible by a square. 
K. Hensep2 f^j. 4^±i^ 6n±l, 8n-l, 8n=t3, 12n-l, lOn-1. 
"Mat. Sbornik (Math. Soc. Moscow), 18, 1896, 519. 
"Proc. Cambr. Phil. Soc, 16, 1911, 9-10. '^Nouv. Ann. Math., (4), 11, 1911, 70-2. 
^^Nieuw Archief voor Wiskunde, (2), 10, 1913, 357-361 (Dutch). 
8«Nouv. Ann. Math., 15, 1856, 130, 236. 
"Exercices d'analyse num^rique, 1859, 91-5, 103-4, 145-6. 
s^Amer. Jour. Math., 1, 1878, 309. saQomptes Rendus Paris, 106, 1888, 1278-81, 1385-6. 
s^Assoc. frang. av. sc, 17, 1888, II, 118-120. 
s^Nyt Tidsskrift for Math., Kjobenhavn, 1891, 2B, 73-82. 
s^Theorie des nombres, 1891, 353-4. "Monatshefte Math. Phys., 7, 1896, 46. 
8'Schriften phys.-okon. Gesell. Konigsberg, 38, 1897, 47. 
"Math. Soc. St. Petersburg, 1899, 53-8 (Russian). 
'"Jour, fur Math., 131, 1906, 265-7; transl. of Math. 6s Phys. Lapok, 14, 1905, 313. 
9iSitzungsber. BerUn Math. Gesell., 11, 1912, 40-50, with Archiv M. P. 
92Zahlentheorie, 1913, 304-5. 
