Chap. XVIII] BeRTRAND's POSTULATE. 435 
E. Landau^''^ indicated errors in rintermediaire des math^maticiens on 
the approximate number of primes ax+h<N. 
*M. Kossler-^^ discussed the relation between Wilson's theorem and the 
number of primes between two limits. 
See Cesaro^^ of Ch. V, Gegenbauer^^ ^f qj^ ^I, and papers 62-81 of 
Ch. XIII. 
Bertrand's Postulate. 
J. Bertrand^®" verified for numbers < 6 000 000 that for any integer 
n>6 there exists at least one prime between n — 2 and n/2. 
P. L. Tchebychef^" obtained Hmits for the sum d{z) of the natural 
logarithms of all primes ^z and deduced Bertrand's postulate that, for 
x>3, there exists a prime between x and 2x — 2. His investigation shows 
that for every e> 1/5 there exists a number ^ such that for every x^^ there 
exists at least one prime between x and (l + e)x. 
A. Desboves,^^^ assuming an unproved theorem of Legendre's,^^ con- 
cluded the existence of at least two primes between any number >6 and 
its double, also between the squares of two consecutive primes; also at 
least p primes between 2n and 2n — k for p and k given and n sufficiently 
large, and hence between a sufficiently large number and its square. 
F. Proth^^^ claimed to prove Bertrand's postulate. 
J. J. Sylvester264 reduced Tchebychef's e to 0.16688. 
L. Oppermann-^^ stated the unproved theorem that if n>l there exists 
at least one prime between n{n — l) and n^, and also between n^ and n(n+l), 
giving a report on the distribution of primes. 
E. C. Catalan^^^ proved that Bertrand's postulate is equivalent to 
n\ n\ 
>a''/5^..7^^ 
where a, . . . , tt denote the primes S n, while a is the number of odd integers 
among [2n/a], [2n/a^],..., h the number among [2n/^], [2n//3^],. . .. He 
noted (p. 31) that if the postulate is applied to 6 — 1 and 6+1, we see the 
existence between 26 and 46 of at least one even number equal to the sum 
of two primes. 
J. J. Sylvester^" reduced Tchebychef's e to 0.092; D. von Sterneck^^^ 
to 0.142. 
24«L'intermediaire des math., 20, 1913, 179; 15, 1908, 148; 16, 1909, 20-1. 
2"Casopis, Prag, 44, 1915, 38-42. 
^^ojour. de I'ecole roy. polyt., cah. 30, tome 17, 1845, 129. 
2"M^m. Ac. Sc. St. Petersbourg, 7, 1854 (1850), 17-33, 27; Oeuvres, 1, 49-70, 63. Jour, de 
Math., 17, 1852, 366-390, 381. Cf. Serret, Cours d'algebre sup^rieure, ed. 2, 2, 1854, 
587; ed. 6, 2, 1910, 226. 
262NOUV. Ami. Math., 14, 1855, 281-295. 
2"Nouv. Corresp. Math., 4, 1878, 236-240. 
2«^Amer. Jour. Math., 4, 1881, 230. 
2660versigt Videnskabs Selsk. Forh., 1882, 169. 
»««M6m. Soc. R. Sc. Liege, (2), 15, 1888 ( = M61anges Math., Ill), 108-110. 
26'Messenger Math., (2), 21, 1891-2, 120. 
268Sitzungsb. Akad. Wiss. Wien, 109, 1900, II a, 1137-58. 
