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January 26th, 2018, 02:30 PM  #1 
Newbie Joined: Jan 2018 From: DeLand, Florida Posts: 1 Thanks: 0  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 
January 26th, 2018, 03:03 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,501 Thanks: 1739 
Any power? 4Â² + 6Â²  3 = 7Â² 
January 26th, 2018, 03:28 PM  #3 
Senior Member Joined: May 2016 From: USA Posts: 1,115 Thanks: 462 
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. 
January 26th, 2018, 03:50 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,501 Thanks: 1739 
Really close misses aren't very common, but 6Â³ + 8Â³ + 1 = 9Â³.

January 28th, 2018, 11:21 AM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,851 Thanks: 1075 Math Focus: Elementary mathematics and beyond  
January 28th, 2018, 01:08 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 19,501 Thanks: 1739 
I don't know a proof.

January 28th, 2018, 01:44 PM  #7 
Senior Member Joined: Aug 2012 Posts: 1,998 Thanks: 569 
$64^3 + 94^3 = 103^3 + 1$ 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=UTF8 Ramanujan found an infinite family of nearmiss cubes related to his studies (decades ahead of his time) of elliptic curves. https://plus.maths.org/content/ramanujan Last edited by Maschke; January 28th, 2018 at 01:50 PM. 

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