Messrs Shah and Wilsons paper 



253 



8. In a later paper* Brun gives a more general formula relating 

 to prime-pairs {p, p) such that p = ap + 2. This formula also 

 involves an undetermined constant k. It is worth pointing out 

 that our method is equally applicable to this and to still more 

 general problems. Suppose, in the first place, that v{n) is the 

 number of expressions of n in the form 



n = ap + hp, 



where pi and p' are primesf. We may suppose without loss of 

 generality that a and h have no common factor. 



The results suggested by our method are as follows. If n has 

 any factor in common with a and h, then 



"<">='' {(log. o-^}' 



and this is true even when n is prime to both a and h, unless one 

 of n, a, b is even|. But if n, a and b are coprime, and one of them 

 even, then 



2J. 



n 



P-i 



ab (log iif \p— 2 



where A is the constant of § 2, and the product is now extended 

 over all odd primes which divide n or a or b. 



* ' Sur les nombres premiers de la forme ap + h\ Archiv for Mathematik, vol. 

 24, 1917, no. 14. 



t We might naturally include powers of primes. 



+ These results are trivial. If n and a have a common factor, it divides hp', 

 and is therefore necessarily p' , which can thus assume but a finite number of values. 

 If n, a, h are all odd, either ^^ ox p' must necessarily be 2. 



