DrKrettin Posted May 20, 2017 Share Posted May 20, 2017 Can anybody prove that eight is the only cube which is one less than a square? Link to comment Share on other sites More sharing options...
imatfaal Posted May 20, 2017 Share Posted May 20, 2017 zero 0^3+1=1^2 Link to comment Share on other sites More sharing options...
DrKrettin Posted May 20, 2017 Author Share Posted May 20, 2017 Can anybody prove that eight is the only positive integer cube which is one less than a square? Link to comment Share on other sites More sharing options...
imatfaal Posted May 20, 2017 Share Posted May 20, 2017 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 2 Link to comment Share on other sites More sharing options...
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