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CraigD · 02-10-2009

Using the general formula
$b^n = \sum_{i=1}^m a_i^n$
where all terms are positive integers, Fermat’s Last Theorem can be written “if $m=2$ , then $n \le 2$ ”.

Considering other values of $m$ appears interesting. For example,
$5^3+4^3+3^3=6^3$
proves that, for $m=3$ , there exist $a_1$ , $a_2$ , $a_3$ and $b$ such that $n = 3$ .

Because $2^n = \sum_{i=1}^{2^n} 1^n$ , we know that there exist $a$ s and $b$ for $m = 2^n$ for any $n$ . It appears possible for $m$ to be much smaller than $2^n$ in many cases, for example
$4^4+4^4+3^4+2^4+2^4=5^4$ and $14^4+9^4+8^4+6^4+4^4=15^4$ for $n=4$ and $m=5$ , and
$11^5+9^5+7^5+6^5+5^5+4^5=12^5$ for $n=5$ and $m=6$

The best my clunky initial searches have managed for $n=6$ is $m=56$ , proven by
$5^6 \, +4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6+4^6$
$+3^6+3^6+3^6+3^6+3^6+3^6+3^6+3^6+3^6+3^6$
$+2^6+2^6+2^6+2^6+2^6+2^6+2^6+2^6$
$+1^6+1^6+1^6+1^6+1^6+1^6+1^6+1^6+1^6+1^6+1^6+1^6+1^6+1^6 = 7^6$

Lots of questions come to mind, such as

Is there always a solution such that $m=n$ ? (proven above for 1, 2 and 3, but not 4, 5, or 6)
If not, is there some other function $f(n) \ge m$ for all $n$ ?
Is there any $m$ such that and $m<n$ ? Fermat’s Last Theorem states that there is not for $m=2$ . Is there some $m$ for which a “generalized FLT” is false? Edit: Yes – see “one question answered”.
Is there a computationally efficient way to generate examples for a given $n$ and $m$ ?
Is there an elementary proof of [1], [2], or [3]? (The only known valid proof or FLT is not elementary)

I’m working on a less clunky approach to [4], and frankly intimidated by [5], and invite you number-crunch and proof-aholics out there to try answering some of these questions, or proposing new ones related to this FMT generalization.

Also, if anyone knows of any literature on this generalization, please share it. Though it seems much too obvious a generalization to be so new, the only mention of it I’ve found is this 2005 arXiv preprint, a 1-page invitation similar to but even briefer than this post.

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