CHAP, xvni] PRIMES IN ARITHMETICAL PROGRESSION. 425 H. Brocard151 gave an incorrect argument by use of Bertrand's postulate that there exists a prime between any two consecutive triangular numbers. G. de Rocquigny162 remarked that it seems true that every multiple of 6 is the difference of two primes of the form 6ft-f-1. Brocard163 verified this property for a wide range of values. L. Kronecker164 remarked that an unnamed writer148 had stated empirically that every even number can be expressed in an infinitude of ways as the difference of two primes. Taking 2 as the number, we conclude that there exist an infinitude of pairs of primes differing by 2. L. Ripert156 verified that every even number < 10000 is a sum of a prime and a power, every odd one except 1549 is such a sum. E. Maillet166 commented on de Polignac's conjecture that every even number is the difference of two primes. E. Maillet157 proved that every odd number < 60000 (or 9-106) is, in default by at most 8 (or 14), the sum of a prune and the double of a prime. PRIMES IN ARITHMETICAL PROGRESSION. E. Waring165 stated that if three primes (the first of which is not 3) are in arithmetical progression, the common difference d is divisible by 6, except for the series 1, 2, 3 and 1, 3, 5. For 5 primes, the first of which is not 5, d is divisible by 30; for 7 primes, the first not 7, d is divisible by 2-S-5-7; for 11 primes, the first not 11, d is divisible by 2-3-5-7-11; and similarly for any prime number of primes in arithmetical progression, a property easily proved. Henop hv pnnt.irmflilxr orirlinar rJ +.r\ a -min-me* -nr^ reach a number divisible by _, J. L. Lagrange166 proved th metical progression, the difference u, ± being 5, d is divisible by 30. He stat 2-3-S-7, unless the first one is 7, and tht^ tive prime terms in a progression whose Qmci-eiiue is nut u E. Mathieu167 proved Waring's statement. M. Cantor168 proved that if P=2»3.. .p is the product of all the primes up to the prime p, there is no arithmetical progression of p primes, no one of which is p, unless the common difference is divisible by P. He conjectured that three successive primes are not in arithmetical progression unless one of them is 3. A. Guibert169 gave a short proof of the theorem stated thus: Let Pi,..., pn be primes ^ 1 in arithmetical progression, where n is odd and >3. Then no prime >1 and ^n is a p^ If n is a prime and is a pi} then t= 1. 161L'intermediaire des math., 4, 1897, 159. Criticism by E. Landau, 20, 1913, 153. ™Ibid., 5, 1898, 268. 1HL'interm<*diaire des math., 6, 1899, 144. ^Vorlesungen tiber Zahlentheorie, 1, 1901, 68. »«L'intenn6diaire des math., 10, ,1903, 217-8. MIbid., 12, 1905, 108. "T/Wd., 13, 1906, 9. ""Meditationes Algebraicae, 1770; ed. 3, 1782, 379. 1MNouv. Mem. Ac. Berlin, annSe 1771,1773,134-7. »7Nouv. Ann. Math., 19,1860, 384r-5. ""Zeitschrift Math. Phye., 6, 1861, 340-3. "•Jour, de Math., (2), 7, 1862, 414-6.