438 HISTORY OF THE THEOEY OF NUMBERS. [CHAP, xvin
0. Meissner291 stated that, if n+1 successive integers w,. m+n are given, we can not in general find another set w^ . . ., wj+n containing a prime m^+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 prune pairs.
G. H. Hardy292 noted that the largest prime dividing a positive integer a; is
Km lim Mm 2[l~(cos{(H)V/z})2n].
r=oo m=oo n«=oo F»0
C. F. Gauss,293 in a manuscript of 1796, stated empirically that the number ir2(x) of integers ^x which are products of two distinct primes, is approximately x log log x/log x.
E. Landau294 proved this result and the generalization
logo:)
-) log* log*
where v,(x) is the number of integers ^x which are products of v distinct primes; also related formulas for v,(x).
Several writers296 gave numerous examples of a sum of consecutive primes equal to an exact power.
E. Landau296 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. Barinaga297 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.
Coblyn298 noted as to prime pairs that, when 4(6p— 2)! is divided by 36p2—l, the remainder is — 6p-3 if 6p — 1 and 6p-f 1 are both primes, zero if both are composite, — 2(6p+l) if only 6p — 1 is prime, and 6p~l if only 6p-|- 1 is prime.
J. Hammond299 gave formulas connecting the number of odd primes <2n, and the number of partitions of 2n into two distinct primes or into two relatively prime composite numbers.
V. Brun300 proved that, however great a is, there exist a successive composite numbers of the form 1 -fw2. There exist a successive primes no two of which differ by 2. He determined a superior limit for the number of prunes <x of a given class.
JWArchiv Math. Phys., 9, 1905, 97.
'"Messenger Math., 35, 1906, 145.
2MCf. F. Klein, Nachrichten Gesell. Wiss. Gfittingcn, 1911, 2fr-32.
™Ibid., 361-381; Handbuch. . .der Primzahlen, I, 1909, 205-211; Bull. Soc. Math. France, 28,
1900, 25-38.
IML'interme"diaire. des math., 18, 1911, 85-6. **Ibid., 20, 1913, 180.
"7L'interme'diaire des math., 20, 1913, 218. "8Soc. Math, de France, C. R. des Stances, 1913, 55.
"9Proo. London Math. Soc., (2), 15, 1916-7, Records of Meetings, Feb. 1916, xxvii. »°°Nyt Tidsskrift for Matematik, B, 27, 1916, 45-58.