CHAP.XVHIJ BEETEANB'S POSTULATE. 435 E. Landau246 indicated errors in rinterme*diaire des math&naticiens on the approximate number of primes ax+b<N. *M. Kossler247 discussed the relation between Wilson's theorem and the number of primes between two limits. See Cesaro64 of Ch. V, Gegenbauer12 of Ch. XI, and papers 62-81 of Ch. XIII. BERTRAND'S POSTULATE. J. Bertrand260 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. Tchebychef261 obtained limits for the sum 6(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-H)x. A. Desboves,262 assuming an unproved theorem of Legendre's,22 concluded 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. Proth263' claimed to prove Bertrand's postulate. J. J. Sylvester264 reduced Tchebychefs c to 0.16688. L. Oppermann265 stated the unproved theorem that if n>l there exists at least one prime between n(n — 1) and n2, and also between n2 and n(n+l), giving a report on the distribution of primes. E. C. Catalan266 proved that Bertrand's postulate is equivalent to where a, . . . , TT denote the primes ^ n, while a is the number of odd integers among [2n/a], [2n/a2],. . ., b the number among [2n/j9], [2n//32], ____ 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. Sylvester267 reduced Tchebychefs e to 0.092; D. von Sterneck268 to 0.142. _ 246L'interm6diaire des math., 20, 1913, 179; 15, 1908, 148; 16, 1909, 20-1. 247Casopis, Prag, 44, 1915, 38-42. 260Jour. de T6cole roy. polyt., cah. 30, tome 17, 1845, 129. 261Me*m. Ac. Sc. St. Pftersbourg, 7, 1854 (1850), 17-33, 27; Oeuvres, 1, 49-70, 63. Jour, de Math., 17, 1852, 366-390, 381. Cf. Serret, Cours d'algebre supe'rieure, ed. 2, 2, 1854, 587; ed. 6, 2, 1910, 226. 262Nouv. Ann. Math., 14, 1855, 281-295. 263Nouv. Corresp. Math., 4, 1878, 236-240. ^Amer. Jour. Math., 4, 1881, 230. 2660versigt Videnskabs Selsk. Forh., 1882, 169. ^MSm. Soc. R. Sc. Liege, (2), 15, 1888 ( = M61angea Math., Ill), 108-110, 287Messenger Math., (2), 21, 1891-2, 120. 268Sitzungsb. Akad. Wiss. Wien, 109, 1900, II a, 1137-58.