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 x4 + 4y4.

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]

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.

x4 + 4y4 factors over Z.

[robert Ihnot]

x4 + 4y4 factors over Z.

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

[Raphie]

x4 + 4y4 factors over Z.

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.