Question

Find all prime numbers p such that 4p2 + 1 and 6p2 + 1 are also primes.

Answer 1

Let P = 2

4P2 + 1 = 4(2)2 + 1 = 17

6P2 + 1 = 6(2)2 + 1 = 25 (not prime)

Let P = 3

4(3)2 + 1 = 37

6(3)2 + 1 = 55 not prime

P = 5

4(5)2 + 1 = 101

6(5)2 + 1 = 151

So 4P2 + 1 and 6P2 + 1 both are prime for P = 5

We know that every square no. is of the form 5 m, 5m+ 1 or 5m + 4

Let take prime P > 5

So P can be 5m + 1 or 5m + 4

Case - I

5m + 1

4P2 + 1 = 4(5m + 1)2 + 1 = 20 k + 5 = 5(4k + 1) (A multiple of 5)

Case - II

5m + 4

6P2 + 1 = 6(5m+ 4)2 + 1 = 30 n + 25 = 5(6n + 5) (A multiple of 5)

So P = 5 is the only solution.