Which Positive Integers Can Be Written as x4 + 4y4 to Form Prime Numbers?

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Discussion Overview

The discussion centers on identifying prime numbers that can be expressed in the form p = x4 + 4y4, where x and y are positive integers. The scope includes mathematical reasoning and factorization techniques related to this expression.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant asks for all prime numbers p that can be expressed as p = x4 + 4y4.
  • Another participant questions whether any such prime numbers exist.
  • Several participants discuss the factorization of x4 + 4y4, with mentions of Gaussian integers and conditions for primes.
  • It is noted that a prime must be congruent to 1 mod 4, and that for the sum to be prime, exactly one of the terms must be divisible by 5, with 5 being a known solution.
  • One participant expresses a desire for a simpler explanation of the concepts being discussed, indicating a lack of understanding of the logical connections made by others.
  • Another participant suggests that the original question may be related to homework, hinting at a more guided approach in private communication.

Areas of Agreement / Disagreement

Participants express differing levels of understanding regarding the factorization and implications of the expression. There is no consensus on the existence of prime numbers fitting the criteria, and the discussion remains unresolved.

Contextual Notes

Some assumptions about the properties of primes and the factorization methods are not fully explored, and there are unresolved mathematical steps regarding the implications of the factors discussed.

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Find all prime numbers p that can be written p = x4 + 4y4 , where x, y are positive
integers.
 
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Do you know that any such prime numbers exist?
 
Factor x4 + 4y4.
 
CRGreathouse said:
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.
 
robert Ihnot said:
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.
 
Petek said:
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.
 
Last edited:
Petek said:
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...Raphie
 
Last edited:
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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