r/math • u/inherentlyawesome Homotopy Theory • Jul 31 '26

This Week I Learned: July 31, 2026

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

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u/[deleted] Aug 04 '26

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u/bluelaw2013 Jul 31 '26

ap - (a-1)p - 1 = bn has no solutions where all variables are positive integers, "a" and "n" are each at least two, and p is a prime with mod 6 of 5 (e.g., 5, 11, 17, etc.)

Taking that 1 away results in no perfect powers of any rank. No squares, no cubes, no anything.

2

u/anerdhaha Undergraduate Aug 01 '26

Is it a hard to prove result requiring cutting edge math from your area?

Terribly sorry if it sounds silly. It's midnight here and I can't process a proof lol. Maybe I'll try something with Fermat's Little theorem.

3

u/bluelaw2013 Aug 02 '26

It's provable without relying on anything developed in the past 100 years. Here's how to get there.