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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 righthand side is $353$, not $353^4\!$.

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 $\displaystyle d=\sqrt[4]{353}$ 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 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,157 Thanks: 732 Math Focus: Physics, mathematical modelling, numerical and computational solutions  
August 25th, 2015, 07:26 AM  #7 
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  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  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,157 Thanks: 732 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
 
September 1st, 2015, 04:09 AM  #9 
Newbie Joined: Jul 2015 From: Singapore Posts: 2 Thanks: 0  

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