Can anybody prove that eight is the only cube which is one less than a square?

zero

 

0^3+1=1^2

Can anybody prove that eight is the only positive integer cube which is one less than a square?

And to answer the question properly (ish) - yes it is provable - Euler proved it in the 18th Century. It is a special case of the (later) Catalan Conjecture. The conjecture states that 3^2-2^3=1 and that this is the only non-trivial solution to x^a-y^b=1 ; this conjecture was only proven in 2003 by Preda Mihailescu

 

Here is an interesting read on the conjecture and its final proof as Mihailescu's Theorem

 

http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf

 

 

 

And here is a copy of Euler's proof and a more modern proof for a=2 and b=3 (page 12 onwards)

 

https://www.mimuw.edu.pl/~zbimar/Catalan.pdf

 

I think I might understand the Latin better than I understand the maths in Eulers