## Prime numbers of given form

Find all prime numbers p that can be written p = x4 + 4y4 , where x, y are positive
integers.
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 Recognitions: Homework Help Science Advisor Do you know that any such prime numbers exist?
 Recognitions: Homework Help Science Advisor Factor x4 + 4y4.

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## Prime numbers of given form

 Quote by CRGreathouse 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.

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 Quote by 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.
x4 + 4y4 factors over Z.

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 Quote by Petek 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.

 Quote by Petek 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
 Recognitions: Gold Member We're just giving hints because the original question probably is homework. I'll send you a PM with more details.

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