436 HISTORY OF THE THEORY OF NUMBERS. [CHAP,
T. J. Stieltjes stated and E. Cahen269 proved that we may take € to be any positive number however small, since B(z) is asymptotic315"6 to z.
H. Brocard270 stated that at least four primes lie between the squares of two consecutive primes, the first being >3. He remarked that this and the similar theorem by Desboves262 can apparently be deduced from Ber-trand's postulate; but this was denied by E. Landau.271
E. Maillet272 proved there is at least one prime between two consecutive squares <9-106 or two consecutive triangular numbers ^9-106.
E. Landau47 (pp. 89-92) proved Bertrand's postulate and hence the existence of a prime between x (excl.) and 2x (incl.) for every x^l.
A. Bonolis273 proved that, if x>13 is a number of p digits and a is the least integer >z/{10(p+l)}, there exist at least a primes between x and [fx—2], which implies Bertrand's postulate. If x> 13 is a number of p digits and ft is the greatest integer <x/(3p—3), there are fewer than P primes from x to [%x—2].
MISCELLANEOUS RESULTS ON PRIMES.
H. F. Scherk280 stated the empirical theorems: Every prime of odd rank (the nth prime 1, 2, 3, 5,... being of rank n) can be composed by addition and subtraction of all the smaller primes, each taken once; thus
13 = 1+2-3-5+7+11= -1+2+3+5-7+11.
Every prime of even rank can be composed similarly, except that the next earlier prime is doubled; thus
17 = 1+2-3-5+7-11+2-13=-1-2+3-5+7-11+2-13.
Marcker281 noted that, if a, b,..., m are the primes between 1 and A and if p is their product, all the primes from A to A2 are given by
and each but once if each numerator is positive and less than its denominator.
0. Terquem282 noted that the prunes <n2 are the odd numbers not included in the arithmetical progressions q2, gf2+2#, <?2+4#,... up to n2, for g = 3, 5,..., n — 1.
H. J. S. Smith283 gave a theoretical method of finding the primes between the xth prime Px and P2z+i, given the first x primes.
C. de Polignac2830 considered the primes ^x in a progression Km+h.
"•Comptes Rendus Paris, 116, 1893, 490; These, 1894, 45; Ann. ficole Normale, (3), 11,1894,
S70L'interm<§diaire des math., 11, 1904, 149.
271/6id., 20, 1913, 177.
272Ib^., 12, 1905, 110-3.
273Atti Ac. Sc. Torino, 47, 1911-12, 576-585.
280Jour. fur Math., 10, 1833, 201.
281J6^., 20, 1840, 350.
S82Nouv. Ann. Math., 5, 1846, 609.
«'Proc. Ashmolean Soc., 3, 1857,128-131; Coll. Math. Papers, 1, 37.
283oComptes Rendus Paris, 54, 1862, 158-9.