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 August 20th, 2015, 04:00 AM #1 Newbie   Joined: Aug 2015 From: Isengard Posts: 7 Thanks: 0 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? August 20th, 2015, 04:25 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,931 Thanks: 2207 Your right-hand side is $353$, not $353^4\!$. Thanks from mick17 August 20th, 2015, 04:43 AM #3 Newbie   Joined: Aug 2015 From: Isengard Posts: 7 Thanks: 0 Yes I noticed my error I misinterpreted the value of d. $\displaystyle d$ in this case is not $\displaystyle 353$. Instead $\displaystyle d=\sqrt{353}$ which is not a natural number. This is solved now. Last edited by skipjack; August 20th, 2015 at 05:29 AM. August 20th, 2015, 05:30 AM #4 Global Moderator   Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms I should mention that Noam Elkies found a counterexample to this conjecture: 2682440^4 + 15365639^4 + 18796760^4 = 20615673^4. August 20th, 2015, 05:33 AM #5 Global Moderator   Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms See also https://oeis.org/A003828 which extends Roger Frye's work in determining the smallest counterexample. August 25th, 2015, 05:29 AM   #6
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 Originally Posted by CRGreathouse 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! August 25th, 2015, 07:26 AM   #7
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 Originally Posted by Benit13 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
$$95800^4+217519^4+414560^4 = 422481^4.$$ August 25th, 2015, 07:47 AM   #8
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 Originally Posted by CRGreathouse 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.  September 1st, 2015, 04:09 AM #9 Newbie   Joined: Jul 2015 From: Singapore Posts: 2 Thanks: 0 Tags integers, positive, proof, solution Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post jiasyuen Number Theory 3 March 19th, 2015 07:55 AM cool012 Algebra 2 December 2nd, 2013 01:21 PM ultramegasuperhyper Number Theory 4 June 5th, 2011 05:07 PM MathematicallyObtuse Algebra 5 January 9th, 2011 09:16 PM hello2413 Number Theory 3 March 15th, 2010 06:22 PM

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