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August 20th, 2015, 04:00 AM   #1
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Why a^4 + b^4 + c^4 = d^4 has no solution when a; b; c; d are positive integers?

I was studying proofs and I came across this proposition which was conjectured by Euler some centuries ago. This is something that's been puzzling me and I'm pretty sure it's because I don't understand the proposition itself.

To me when it is said that

"$\displaystyle a^4 + b^4 + c^4 = d^4$ has no solution when a; b; c; d are positive integers"

I understand that a, b, c and can't hold natural numbers and keep the equality. However what if you have

$\displaystyle a=2$

$\displaystyle b=3$

$\displaystyle c=4$

$\displaystyle d= 353$

$\displaystyle 2^4 + 3^4 + 4^4 = 353$

the right hand side of the equality yields a natural number (which is not 0)

And the values of a, b, c are all natural numbers (again no 0).

So this means there are solutions when a, b ,c and d are positive integers?

Can anybody clearly explain what is exactly meant by this proposition?
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August 20th, 2015, 04:25 AM   #2
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Your right-hand side is $353$, not $353^4\!$.
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August 20th, 2015, 04:43 AM   #3
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Yes I noticed my error I misinterpreted the value of d.

$\displaystyle d$ in this case is not $\displaystyle 353$.

Instead $\displaystyle d=\sqrt[4]{353}$ which is not a natural number.

This is solved now.

Last edited by skipjack; August 20th, 2015 at 05:29 AM.
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August 20th, 2015, 05:30 AM   #4
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I should mention that Noam Elkies found a counterexample to this conjecture:

2682440^4 + 15365639^4 + 18796760^4 = 20615673^4.
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August 20th, 2015, 05:33 AM   #5
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See also
https://oeis.org/A003828
which extends Roger Frye's work in determining the smallest counterexample.
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August 25th, 2015, 05:29 AM   #6
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Quote:
Originally Posted by CRGreathouse View Post
I should mention that Noam Elkies found a counterexample to this conjecture:

2682440^4 + 15365639^4 + 18796760^4 = 20615673^4.
That's nuts! Mathematicians are amazing!
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August 25th, 2015, 07:26 AM   #7
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Originally Posted by Benit13 View Post
That's nuts! Mathematicians are amazing!
It's even crazier than it might seem -- he didn't just brute force it, he found a connection with a certain class of elliptic curves which led him to that solution. Frye later found a good method for finding the smallest solution
$$
95800^4+217519^4+414560^4 = 422481^4.
$$
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August 25th, 2015, 07:47 AM   #8
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Quote:
Originally Posted by CRGreathouse View Post
It's even crazier than it might seem -- he didn't just brute force it, he found a connection with a certain class of elliptic curves which led him to that solution. Frye later found a good method for finding the smallest solution
$$
95800^4+217519^4+414560^4 = 422481^4.
$$
I've downloaded the paper from the link you put up and I'm going to try and follow it through.
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September 1st, 2015, 04:09 AM   #9
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(Euler Quartic Conjecture)
Project Euler Quartic Conjecture Гипотеза Эйлера (Euler Quartic Conjecture)
http://thales.math.uqam.ca/~rowland/...%5E4=D%5E4.pdf
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