How many solutions (a,c) are there such that both numbers are positive integers?

Please comment upon the unexpected result.

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since 15226= 2*7613 and 7613 is a prime number  -    let's solve for a general case: 

(c+a) *(c-a) =2*p , p an odd prime

Solving for a & c there are 2 possible integer factorizations :

   c+a =2p   c-a=1    or

 c+a=p ,    c-a=2

In both sets of equation there are no   integer  solutions:

c+a and c-a  must be of the same parity, i.e. both odd or both even.

see my "spoiler" post

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Edited on March 1, 2018, 4:15 am

Posted by Ady TZIDON on 2018-02-28 05:28:07