My Math Forum Why a^4 + b^4 + c^4 = d^4 has no solution when a; b; c; d are positive integers?

 Abstract Algebra Abstract Algebra Math Forum

 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 $\displaystyle d=\sqrt[4]{353}$ 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
Quote:
 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
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
Quote:
 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
Senior Member

Joined: Apr 2014
From: Glasgow

Posts: 2,157
Thanks: 732

Math Focus: Physics, mathematical modelling, numerical and computational solutions
Quote:
 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 $$95800^4+217519^4+414560^4 = 422481^4.$$

 September 1st, 2015, 04:09 AM #9 Newbie   Joined: Jul 2015 From: Singapore Posts: 2 Thanks: 0

 Tags integers, positive, proof, solution

 Thread Tools Display Modes Linear 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

 Contact - Home - Forums - Cryptocurrency Forum - Top