I've realized that there are maybe no positive integers $(a, b)$ that $a^4 + b^4$ is a perfect square, because I tested for $a, b \le 10000$ and cannot find any solution.
I think that's weird, and how to prove it?
Asked
Active
Viewed 76 times
0
square1001
- 565
-
1Fermat proved this by "infinite descent". See texts on number theory. – Angina Seng Aug 14 '17 at 11:16
-
I don't have a text of number theory. I'm not a university student. (and I'm a middle school student and only know high-school level maths) If you have a link, can you provide? – square1001 Aug 14 '17 at 11:25
-
2Googling "fermat's last theorem power of 4" gave me this as the first result and this as the third. – Arthur Aug 14 '17 at 11:26
-
1Okay it helps - because when I googled it I only found the result of proof of $x^4-y^4=z^2$. – square1001 Aug 14 '17 at 11:30
-
One pretty proof is via Fibonacci's Lost Theorem. – Bill Dubuque Aug 14 '17 at 17:21
-
Perhaps someone could consider making one (or more) of their comments into an answer, so the OP can accept one, and this post can move off the "Unanswered" queue. Thanks! – Kieren MacMillan Sep 03 '17 at 20:11