Mr.Math Group Title Find all integers a,b for which $$a^4+4b^4$$ is a prime. 2 years ago 2 years ago

1. FoolForMath Group Title

a=b=1 is the only possible solution.

2. Mr.Math Group Title

Could you prove your claim? :-P

3. Hero Group Title

Yeah FFM

4. FoolForMath Group Title

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

5. Mr.Math Group Title

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

6. Hero Group Title

FFM, stop stalling and post the proof :P

7. FoolForMath Group Title

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)

8. Mr.Math Group Title

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

9. FoolForMath Group Title

Haha, how old are you man? :D

10. Mr.Math Group Title

Very old.

11. Hero Group Title

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

12. FoolForMath Group Title

Hero, I really wonder that myself ...