My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Thanks Tree2Thanks
  • 2 Post By Maschke
Reply
 
LinkBack Thread Tools Display Modes
January 26th, 2018, 02:30 PM   #1
Newbie
 
Joined: Jan 2018
From: DeLand, Florida

Posts: 1
Thanks: 0

Question Fermat Theorem Question

In Fermat's Last Theorem ... do the numbers to the left of the equal sign have to be different? I'm looking for close solutions ... and have (after several years)found two systems which (I think) predicts them ... possibly to infinity.
For example 4^3 + 4^3 -3 = 5^3 or 64 + 64 - 3 = 125
This is the ONLY equation I can find which is "off by three" AT ANY POWER.
But if the first two numbers must be different, even this is not a valid example.
Thank you,
OldDumbGuy
OldDumbGuy is offline  
 
January 26th, 2018, 03:03 PM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 19,190
Thanks: 1649

Any power? 4² + 6² - 3 = 7²
skipjack is offline  
January 26th, 2018, 03:28 PM   #3
Senior Member
 
Joined: May 2016
From: USA

Posts: 1,053
Thanks: 431

The numbers may be the same, but

$5^3 + 5^3 = 250 \text { and } 6^3 = 216 \implies 6^3 = 5^3 + 5^3 - 34.$

And 3 does not equal 34.

So I do not understand what you are saying.

In general

$a^3 + a^3 - (a^3 - 3a^2 - 3a - 1) = a^3 + 3a^2 + 3a + 1 = (a + 1)^3.$

$\therefore a = 4 \implies 4^3 - 3 * 4^2 - 3 * 4 - 1 = 64 - 48 - 12 - 1 = 3 \implies$

$4^3 + 4^3 - 3 = 5^3$ but that is not a general result.
JeffM1 is online now  
January 26th, 2018, 03:50 PM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 19,190
Thanks: 1649

Really close misses aren't very common, but 6³ + 8³ + 1 = 9³.
skipjack is offline  
January 28th, 2018, 11:21 AM   #5
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,825
Thanks: 1051

Math Focus: Elementary mathematics and beyond
Quote:
Originally Posted by skipjack View Post
Really close misses aren't very common...
Can you prove that? I'm curious as to how that would be done.
greg1313 is offline  
January 28th, 2018, 01:08 PM   #6
Global Moderator
 
Joined: Dec 2006

Posts: 19,190
Thanks: 1649

I don't know a proof.
skipjack is offline  
January 28th, 2018, 01:44 PM   #7
Senior Member
 
Joined: Aug 2012

Posts: 1,960
Thanks: 547

$64^3 + 94^3 = 103^3 + 1$

A Google search for "Fermat near misses" turns up many links on the subject for anyone interested. They're not all that rare but I didn't dig into the details of these articles.

https://www.google.com/search?q=ferm...hrome&ie=UTF-8

Ramanujan found an infinite family of near-miss cubes related to his studies (decades ahead of his time) of elliptic curves.

https://plus.maths.org/content/ramanujan
Thanks from greg1313 and Joppy

Last edited by Maschke; January 28th, 2018 at 01:50 PM.
Maschke is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
fermat, question, theorem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Fermat with binomial theorem question Richard Fenton Number Theory 3 January 13th, 2015 08:29 AM
Fermat's little theorem ( stuck on completing a question) fe phi fo Number Theory 3 May 6th, 2012 11:17 AM
More Fermat's Last Theorem. theomoaner Number Theory 29 November 26th, 2011 10:23 PM
Brief : Fermat's last theorem rnck Number Theory 9 August 24th, 2011 04:58 AM
fermat's last theorem???? smslca Number Theory 4 September 14th, 2010 08:00 PM





Copyright © 2018 My Math Forum. All rights reserved.