Mr.Math
Find all integers a,b for which \(a^4+4b^4\) is a prime.
Delete
Share
This Question is Closed
FoolForMath
Best Response
You've already chosen the best response.
7
a=b=1 is the only possible solution.
Mr.Math
Best Response
You've already chosen the best response.
1
Could you prove your claim? :-P
Hero
Best Response
You've already chosen the best response.
0
Yeah FFM
FoolForMath
Best Response
You've already chosen the best response.
7
Yes I definitely can ... why you don't believe me MR.Math? :(
Mr.Math
Best Response
You've already chosen the best response.
1
lol, I do believe you! Where did I say that I don't? :-(
Hero
Best Response
You've already chosen the best response.
0
FFM, stop stalling and post the proof :P
FoolForMath
Best Response
You've already chosen the best response.
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)
Mr.Math
Best Response
You've already chosen the best response.
1
Awesome! I've always believed in you son :-)
FoolForMath
Best Response
You've already chosen the best response.
7
Haha, how old are you man? :D
Mr.Math
Best Response
You've already chosen the best response.
1
Very old.
Hero
Best Response
You've already chosen the best response.
0
FFM, how do you come up with such approaches to proofs?
FoolForMath
Best Response
You've already chosen the best response.
7
Hero, I really wonder that myself ...