Chap. XVIII] PRIMES IN ARITHMETICAL PROGRESSION. 425 
H. Brocard^^^ gave an incorrect argument by use of Bertrand's postulate 
that there exists a prime between any two consecutive triangular numbers. 
G. de Rocquigny^^^ remarked that it seems true that every multiple of 
6 is the difference of two primes of the form 6n+l. 
Brocard^^^ verified this property for a wide range of values. 
L. Kronecker^^^ remarked that an unnamed writer'^^ had stated empiri- 
cally 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. Ripert^^^ 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. Maillet^^^ commented on de Polignac's conjecture that every even 
number is the difference of two primes. 
E. Maillet^" proved that every odd number < 60000 (or 9-10^) is, in 
default by at most 8 (or 14), the sum of a prime and the double of a prime. 
Primes in Arithmetical Progression. 
E. Waring^^^ 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 
2-3-5-7; 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. Hence by continually adding d to a prime, we 
reach a number divisible by 3, 5, ... , unless d is divisible by 3, 5, . . . . 
J. L. Lagrange ^^"^ proved that if 3 primes, no one being 3, are in arith- 
metical progression, the difference d is divisible by 6; for 5 primes, no one 
being 5, d is divisible by 30. He stated that for 7 primes, d is divisible by 
2-3-5'7, unless the first one is 7, and then there are not more than 7 consecu- 
tive prime terms in a progression whose difference is not divisible by 2-3 -5 -7. 
E. Mathieu^^^ proved Waring's statement. 
M. Cantor^ ®^ 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 conjec- 
tured that three successive primes are not in arithmetical progression unless 
one of them is 3. 
A. Guibert^®^ 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 Pi. If n is a prime and is a pi, then i = l. 
i"L'intermediaire des math., 4, 1897, 159. Criticism by E. Landau, 20, 1913, 153. 
is^/feid., 5, 1898, 268. i^L'intermediaire des math., 6, 1899, 144. 
i64Vorlesungen iiber Zahlentheorie, 1, 1901, 68. 
i«L'interm6diaire des math., 10, 1903, 217-8. i"/6td., 12, 1905, 108. 
^"Ibid., 13, 1906, 9. i«*Meditationes Algebraicae, 1770; ed. 3, 1782, 379. 
i««Nouv. M6m. Ac. Berlin, ann^e 1771, 1773, 134-7. »"Nouv. Ann. Math., 19, 1860, 384-5. 
"sZeitschrift Math. Phys., 6, 1861, 340-3. 
"»Jour. de Math., (2), 7, 1862, 414-6. 
