Science Forums 2

Turtle · 02-10-2009

Quote:

Originally Posted by CraigD

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

...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.

Interesting questions C. Just a couple observations.

On $n=6$ is $m=56$ , m & n are both multiples of Perfect Numbers.

Your generalized approach put me in mind of Euler & his proofs on figurate numbers, and that Squares belong to that group. Maybe there's something in that you can use here, since Squares are your exception?

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