Messrs Shall and Wilson's paper 251 



6. A modification of our method enables us to attack a closely 

 related problem, that of the existence of pairs of primes differing 

 by a constant even number k. 



We have 



2 A (n) A (n + k) r^''+'' = J- f '" \f{re^^) \ ' e'^'^ cW, 



where f(x) is the same function as in § 1, and r is positive and less 

 than unity. We divide the range of integration into a number of 

 small arcs, correlated in an appropriate manner with a certain 

 number of the points e"P''^"J, and approximate to {/(j'e'")!^ on each 

 arc by means of the formula (2-8). The result thus suggested is 

 that 



^A{n)A (n + k) r- ^ ^^~ U (^ £ ^) , 



where A has the same meaning as in § 2 and p is an odd prime 

 divisor of k. From this it would follow that 



(6-1) S A (v) A (v + k) - 2AnU P^) ; 



and that, if A^^. (?2) is the number of prime pairs less than /?, whose 

 difference is k, then 



T.r / N 2ylri „ /p — 1\ 



(6-2) ^V'«~(i^^,n(P-2). 



This formula is of exactly the same form as (1), except that p is 

 now a factor of k and not of n. In particular we should have 



,_ . , 2An 

 (6-3) ^^(»)~(lo-g»r 



and 



(6-4) ^^"<»>~(Ttg-LV 



We should therefore conclude that there are about two pairs of 

 primes differing by 6 to every pair differing by 2. This conclusion 

 is easily verified. In fact the numbers of pairs differing by 2, below 

 the limits* 



100, 500, 1000, 2000, 3000, 4000, 5000, 

 are 



9, 24, 35, 61, 81, 103, 125; 



while the numbers of pairs differing by 6 are 



16, 47, 73, 125, 168, 201, 241. 



* To be precise, the numbers of pairs ( j), p') such that p' =p + 2 and p' does not 

 exceed the limit in question. 



