436 History of the Theory of Numbers. [Chap, xvill 
T. J. Stieltjes stated and E. Cahen'^^^ proved that we may take e to be 
any positive number however small, since d{z) is asjTnptotic^^^"® to z. 
H. Brocard"" 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 Desboves-^- can apparently be deduced from Ber- 
trand's postulate; but this was denied by E. Landau. ^^^ 
E. ^laillet^^- proved there is at least one prime between two consecutive 
squares <9-10^ or two consecutive triangular numbers ^9-10^. 
E. Landau"*^ (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. Bonolis-"^ proved that, if x>13 is a number of p digits and a is the 
least integer >x/{lO{p-{-l)], there exist at least a primes between x and 
[-|x — 2], which implies Bertrand's postulate. If x>l3 is a number of 
p digits and /3 is the greatest integer <x/(3p— 3), there are fewer than 
^ primes from a: to [-|x — 2]. 
Miscellaneous Results on Primes. 
H. F. Scherk^^*^ 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. 
Marcker^^^ noted that, if a, 6, . . . , m are the primes between 1 and A 
and if p is their product, all the primes from A to A^ are given by 
K^?+ ■+^>+»)' 
and each but once if each numerator is positive and less than its denominator. 
0. Terquem"^- noted that the primes <rr are the odd numbers not 
included in the arithmetical progressions q^, q^-\-2q, 5^+4g, . . . up to n^, 
for g = 3, 5, . . ., n — 1. 
H. J. S. Smith-^^ gave a theoretical method of finding the primes between 
the xth prime P^ and P'^x+i, given the first x primes. 
C. de Polignac^^^" considered the primes ^x in a progression Km-\-h. 
""Comptes Rendus Paris, 116, 1893, 490; These, 1894, 45; Ann. ficole Normale, (3), 11, 1894. 
""L'intermddiaire des math., 11, 1904, 149. 
'■'/bMf., 20, 1913, 177. 
"'/6trf., 12, 1905, 110-3. 
*"Atti Ac. Sc. Torino, 47, 1911-12, 576-585. 
"ojour. fur Math., 10, 1833, 201. 
"'/twi., 20, 1840, 350. 
"'Nouv. Ann. Math., 5, 1846, 609. 
»8»Proc. Ashmolean Soc, 3, 1857, 128-131; Coll. Math. Papers, 1, 37. 
""K>)mptes Rendus Paris, 54, 1862, 158-9. 
