Talk:Fermat–Catalan conjecture

Beal conjecture

We can bump "as of 2008" to "as of 2013" or "as of today". — Preceding unsigned comment added by EdPeggJr (talk • contribs) 12:16, 7 June 2013 (UTC)

When mentioning Falting's theorem, one should say about the Beal's conjecture too. I is somehow important. — Preceding unsigned comment added by 223.27.210.130 (talk) 07:19, 5 November 2011 (UTC)

Note

The phrase "coprime integers" has more than one meaning if three whole numbers are involved. — Preceding unsigned comment added by 2A00:23C4:7C87:4F00:15D4:7730:972E:49E (talk) 10:01, 8 July 2020 (UTC)

New unpublished solutions

In special:diff/1276665134, 150.224.2.218 (presumably Henry Glover) posted the following:

--New by Henry Glover (unpublished, annoyed :) mathematician)--

$1011^{4}+1348^{4}=337^{5}\;$

$33^{5}+66^{5}=33^{6}\;$

$756^{3}+945^{3}=189^{4}\;$

$194^{4}+291^{4}=97^{5}\;$

$70^{3}+105^{3}=35^{4}\;$

(...I have 40 more love :)

I'm posting here because even if I posted my solution on public websites, it would never get noticed (I have tried, hence annoyed). If a journal agrees to publish me, then I'll publish this proof and many more proofs.

This kind of thing probably needs a source to add to the article, ideally published but even an arxiv preprint or similar might do. But at the very least we can mention the edit on the talk page here.

However, at WT:WPM § Fermat–Catalan conjecture (special:diff/1276686240) XOR'easter points out:

The condition on the exponents is met, but the triples aren't coprime. 70, 105, and 35 have common factors of 5 and 7; 194 and 291 are multiples of 97; 756 and 945 are multiples of 189; 66 is twice 33; 1011 and 1348 are multiples of 337.

–jacobolus (t) 07:21, 20 February 2025 (UTC)

Indeed not just not coprime but specifically generatable by the following (completely uninteresting) procedure: for any positive integers k, x, y, let $z=x^{k}+y^{k}$. Then setting $a=xz,b=yz,c=z$ we have $a^{k}+b^{k}=c^{k+1}$. --JBL (talk) 18:22, 20 February 2025 (UTC)

I missed the coprime condition upon first examination. Bubba73 ^{You talkin' to me?} 23:19, 20 February 2025 (UTC)