Mr.Math
  • Mr.Math
Find all integers a,b for which \(a^4+4b^4\) is a prime.
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
a=b=1 is the only possible solution.
Mr.Math
  • Mr.Math
Could you prove your claim? :-P
Hero
  • Hero
Yeah FFM

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anonymous
  • anonymous
Yes I definitely can ... why you don't believe me MR.Math? :(
Mr.Math
  • Mr.Math
lol, I do believe you! Where did I say that I don't? :-(
Hero
  • Hero
FFM, stop stalling and post the proof :P
anonymous
  • anonymous
hehe, okay here it goes, \[ a^4+4b^4= ((a+b)^2+b^2) \times ((a-b)^2+b^2)) \] Now, \( ((a+b)^2+b^2) \gt 1\) (always) so for \( a^4+4b^4 \) to be prime \((a-b)^2+b^2) =1\) and this can only happen when \(a=b=1\) (QED)
Mr.Math
  • Mr.Math
Awesome! I've always believed in you son :-)
anonymous
  • anonymous
Haha, how old are you man? :D
Mr.Math
  • Mr.Math
Very old.
Hero
  • Hero
FFM, how do you come up with such approaches to proofs?
anonymous
  • anonymous
Hero, I really wonder that myself ...

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