Determine all possible values of a prime number p such that each of
6p² + 1 and
4p² + 1 is a prime number.

Ans: 5 .    

Solution:  Consider the numbers a1= 4p^2+1 and a2=6p^2+1 modulo 5.

It is easy to show that one of them is always 0 mod 5 for any k if the prime is 5n+k, and k is not 0.

The only prime number with k=0   is 5.

Then a1=101 and a2=151, both primes. 

     

 

 

 

 

 

 

 

   

Edited on September 27, 2016, 8:26 am

Posted by Ady TZIDON on 2016-09-27 08:17:42