Thread: Equal Sums of Like Powers

1. Equal Sums of Like Powers

2. Interesting.

It seems to be a little old - May 6, 2001. Also, it seems to be focused on small solutions - large solutions can be useful, too.

Someone on MHF has an example of equal sums of like powers in their signature, but I forgot who. I looked around and found Swlabr has $3987^{12}+4365^{12}=4472^{12}$, but that's not the one I was thinking of. By the way, is Swlabr's counter-example to FLT some kind of inside joke?

- Hollywood

3. Originally Posted by hollywood
Interesting.

It seems to be a little old - May 6, 2001. Also, it seems to be focused on small solutions - large solutions can be useful, too.

Someone on MHF has an example of equal sums of like powers in their signature, but I forgot who. I looked around and found Swlabr has $3987^{12}+4365^{12}=4472^{12}$, but that's not the one I was thinking of. By the way, is Swlabr's counter-example to FLT some kind of inside joke?

- Hollywood
You're probably thinking of Opalg's $\color{blue}95800^4 + 217519^4 + 414560^4 = 422481^4$.

I'm also curious to know if Swlabr's is an inside joke.

4. Yes, that's it.

The site in question seems to be looking only for cases with the same number of terms on both sides. Of course, I suppose you can set some of the numbers to zero. Also, they are looking for cases where the equation is true for multiple exponents. Here's an interesting example from the site:

$975^2+224368^2+300495^2+366448^2=37648^2+202575^2+ 337168^2+344655^2$
$975^3+224368^3+300495^3+366448^3=37648^3+202575^3+ 337168^3+344655^3$
$975^4+224368^4+300495^4+366448^4=37648^4+202575^4+ 337168^4+344655^4$

- Hollywood

5. Hollywood

Originally Posted by hollywood
Yes, that's it.

The site in question seems to be looking only for cases with the same number of terms on both sides. Of course, I suppose you can set some of the numbers to zero. Also, they are looking for cases where the equation is true for multiple exponents. Here's an interesting example from the site:

$975^2+224368^2+300495^2+366448^2=37648^2+202575^2+ 337168^2+344655^2$
$975^3+224368^3+300495^3+366448^3=37648^3+202575^3+ 337168^3+344655^3$
$975^4+224368^4+300495^4+366448^4=37648^4+202575^4+ 337168^4+344655^4$

- Hollywood
That example is part of my next article.

I believe there's much more to explore with multigrades which can keep mathematicians busy for a lifetime (besides what I already covered in my article).

6. Originally Posted by hollywood
Interesting.
I looked around and found Swlabr has $3987^{12}+4365^{12}=4472^{12}$, but that's not the one I was thinking of. By the way, is Swlabr's counter-example to FLT some kind of inside joke?

- Hollywood
I think that this "counter-example" may have appeared in a Simpsons episode.

Fermat's Last Theorem in fiction - Wikipedia, the free encyclopedia