LPS conjecture?

Number Theory Level 3

If there exist six natural numbers, a, b, c, d, e, and f that satisfy the equation a 6 + b 6 + c 6 + d 6 + e 6 = f 6 , a^6 + b^6 + c^6 + d^6 + e^6 = f^6, at least how many of them are divisible by 7?


The answer is 4.

Nicholas Stearns by Nicholas Stearns , 25, USA

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1 solution

Michael Mendrin
Aug 18, 2016

Okay, this is nice, being that M o d ( n 6 , 7 ) Mod({n}^{6},7) is always 1 1 , except when n n is a multiple of 7 7 . Hence, IF any such natural numbers a , b , c , d , e , f a, b, c, d, e, f exist that satisfies the equation as given, 4 4 of them have to be a multiple of 7 7 . However, no such set of six natural numbers exist that can satisfy this equation!

Maybe this is a trick question?

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It's actually not known whether a solution exists or not. Perhaps I should've stated it as "If a solution exists..."?

Nicholas Stearns - 4 years, 10 months ago

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yeah, something like that---"if", or "should natural numbers exist", etc? Often, when trying to find a solution, we do consider steps such as finding how how many need to be multiples of 7 7 , for example, which can end up being part of the proof why no such set of integers can be found.

Michael Mendrin - 4 years, 10 months ago

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