438 History of the Theory of Numbers. [Chap, xviii 
0. !Meissner^^^ stated that, if n+1 successive integers m,. . ., m+n are 
given, we can not in general find another set mi,. . ., mi-\-n containing a 
prime 7ni-\-v corresponding to every prime m+v of the first set. But 
for n = 2, it is supposed true that there exist an infinitude of prime pairs. 
G. H. Hardy ^^^ noted that the largest prime dividing a positive integer x is 
m 
lim lim Hm i:[l-{cos{{v\yir/x]n 
r=oo TO=oo n = oo v=0 
C. F. Gauss,^^^ in a manuscript of 1796, stated empirically that the 
number T2ix) of integers ^x which are products of two distinct primes, is 
approximately x log log x/log x. 
E. Landau^^^ proved this result and the generalization 
7r,(x) = 
1 a^qoglogx) "-^ ''-^i-- 1 ^"-2' 
{v — l)\ log a; 
^ ^ | x(log log x) " ^\ 
1 log X J 
where irXx) is the number of integers ^ x which are products of v distinct 
primes; also related formulas for ir^x). 
Several writers-^^ gave numerous examples of a sum of consecutive primes 
equal to an exact power. 
E. Landau^^^ proved that the probability that a number of n digits be a 
prime, when n increases indefinitely, is asymptotically equal to l/(n log 10). 
J. Barinaga^^^ expressed the sum of the first n primes as a product of 
distinct primes for n = 3, 7, 9, 11, 12, 16, 22, 27, 28, and asked if there is a 
general law. 
Coblyn^^^ noted as to prime pairs that, when 4(6p — 2)! is divided by 
36p2 — 1, the remainder is —6p — S if 6p — 1 and 6p-i-l are both primes, 
zero if both are composite, — 2(6p+l) if only Qp — 1 is prime, and Qp — 1 if 
only 6p4-l is prime. 
J. Hammond^^^ gave formulas connecting the number of odd primes 
<2n, and the number of partitions of 2n into two distinct prunes or into 
two relatively prime composite numbers. 
V. Brun^^*^ proved that, however great a is, there exist a successive com- 
posite numbers of the form l-\-u^. There exist a successive primes no two 
of which differ by 2. He determined a superior limit for the number of 
primes < x of a given class. 
"•Archiv Math. Phys., 9, 1905, 97. 
'^^MessenRer Math., 35, 1906, 145. 
"Cf. F. Klein, Nachrichten Ge.sell. Wiss. Gottingen, 1911, 26-32. 
'^Ibid., 361-381; Handbuch. . .der Primzahlen, I, 1909, 205-211; Bull. Soc. Math. France, 28, 
1900, 25-38. 
""L'interm^diaire des math., 18, 1911, 85-6. 
^*^Ibid., 20, 1913, 180. 
"'L'intermddiaire des math., 20, 1913, 218. 
"«Soc. Math, de France, C. R. des Stances, 1913, 55. 
»»Proo. London Math. Soc, (2), 15, 1916-7, Records of Meetings, Feb. 1916, xxvii. 
"""Nyt Tidsskrift for Matematik, B, 27, 1916, 45-58. 
