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Two Primes From Prime (Posted on 2016-09-27) Difficulty: 3 of 5
Determine all possible values of a prime number p such that each of
6p2 + 1 and
4p2 + 1 is a prime number.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (2 votes)

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Solution solution | Comment 1 of 3

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

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