Prime numbers of given form

sachinism

Find all prime numbers p that can be written p = x⁴ + 4y⁴ , where x, y are positive
integers.

Tide

Do you know that any such prime numbers exist?

CRGreathouse

Factor x⁴ + 4y⁴.

robert Ihnot

Factor, what, over the gaussian integers? In that case we know that a prime must be congruent to 1 mod 4. Also, other than 5, which is a solution, exactly one of the terms is divisible by 5, for the sum to be prime.

Petek

x⁴ + 4y⁴ factors over Z.

robert Ihnot

Shiver me timbers, I do see that is correct! So all we'd have to show is that the smaller factor exceeds 1.

Raphie

Would anyone care to explain this in overly simplistic terms for a mathematically oriented, but untrained, layman such as myself? I mean, I know what "Z" is and I know what "factors" and "factorizations" are (at least simplistically speaking...), and I even am familiar with Gaussian versus, say, Eisenstein integers (again, simplistically speaking...), but the rest rather escapes me. In other words I am not following the logical train of thought that is obvious to other posters upon this thread...

Best,
Raphie

Petek

We're just giving hints because the original question probably is homework. I'll send you a PM with more details.