Prove that a⁴+4 is never prime for any a bigger than 1.

For two values that differ by two, their values in the expression always have a common factor:

[(n+1)^4-4]-[(n-1)^4+4]=8n^2+8n=8n(n^2+1)

This n^2+1 is a common factor:

13^4+4=5*29*197
15^4+4=197*259
14^2+1=197

So since a=2 and a=3 are not prime, not greater values of a can yield primes.

Posted by Jer on 2016-07-13 10:20:51