Question

Given the pair of prime numbers p and 8p² + 1, find p. Number p​

Answer 1

Step-by-step explanation:

Prime numbers different from 2,3,5 are either of the form 6k+1 or of the form 6k+5. Insert p=6k+1 in 8p2+1 and conclude. Then insert p=6k+5 in 8p2+1 and conclude. Some cases are still there to check them so check them.

Some kind of converse is also true:

If both 8p2+1 and 8p2−1 are primes then they are twin primes and the sum of two twin primes is always divisible by 12 if they are not the pair (3,5).

So we have 12|16p2 which implies 3|4p2. So p=3.