Equal Sums of Like Powers

• May 26th 2010, 04:38 PM
wonderboy1953
Equal Sums of Like Powers
• May 27th 2010, 08:45 AM
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 \$\displaystyle 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
• May 27th 2010, 10:44 AM
undefined
Quote:

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 \$\displaystyle 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 \$\displaystyle \color{blue}95800^4 + 217519^4 + 414560^4 = 422481^4\$.

I'm also curious to know if Swlabr's is an inside joke.
• May 27th 2010, 04:21 PM
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:

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

- Hollywood
• May 28th 2010, 07:32 AM
wonderboy1953
Hollywood
Quote:

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:

\$\displaystyle 975^2+224368^2+300495^2+366448^2=37648^2+202575^2+ 337168^2+344655^2\$
\$\displaystyle 975^3+224368^3+300495^3+366448^3=37648^3+202575^3+ 337168^3+344655^3\$
\$\displaystyle 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).
• May 29th 2010, 02:55 AM
roninpro
Quote:

Originally Posted by hollywood
Interesting.
I looked around and found Swlabr has \$\displaystyle 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