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Go one, two, three (Posted on 2021-01-09) Difficulty: 3 of 5
Solve in natural numbers a2=2b3c+1

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 3.0000 (1 votes)

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re(2): Solution Comment 6 of 6 |
(In reply to re: Solution by Steve Herman)

A piece of math trivia is that 8 and 9 are the only consecutive perfect powers.  More formally this is written in the form of Catalan's Conjecture:

The equation x*a - y^b = 1 with x,y>0 and a,b>1 
has exactly one solution for which all a,b,x,y are integers.
That solution is 3^2 - 2^3 = 1
A proof of the conjecture was accepted around 2003-4.  More specifically the subcase where the bases are limited 2 and 3 was proven before Catalan formally published his conjecture.

https://en.wikipedia.org/wiki/Catalan's_conjecture
https://mathworld.wolfram.com/CatalansConjecture.html

I probably should have included information this in my solution, or if I was feeling ambitious proven the bases 2 and 3 subcase.

  Posted by Brian Smith on 2021-01-17 00:38:50
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