Question

Find all integers a,b for which \(a^4+4b^4\) is a prime.

Answer 1

a=b=1 is the only possible solution.

Answer 2

Could you prove your claim? :-P

Answer 3

Yeah FFM

Answer 4

Yes I definitely can ... why you don't believe me MR.Math? :(

Answer 5

lol, I do believe you! Where did I say that I don't? :-(

Answer 6

FFM, stop stalling and post the proof :P

Answer 7

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)

Answer 8

Awesome! I've always believed in you son :-)

Answer 9

Haha, how old are you man? :D

Answer 10

Very old.

Answer 11

FFM, how do you come up with such approaches to proofs?

Answer 12

Hero, I really wonder that myself ...