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- Mar 2nd 2009, 02:18 AM #1

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## x^3 = y^2 + 1

Hey guys, I was wondering do any of you have any insight into the problem mentioned in the title. Here it is again for good measure:

with

I reckon there's only 1 sol'n, i.e. x = 3 and y = 5

However, despite all my attempts to prove this, nothing seems to work. Any thoughts?

- Mar 2nd 2009, 02:52 AM #2

- Mar 2nd 2009, 09:02 AM #3

- Mar 2nd 2009, 12:58 PM #4
Catalan's conjecture states that and are the only consecutive natural numbers that are both powers. This result was first proved by Mihăilescu in 2002 (more than 150 years after Catalan first stated it). It follows as a special case that the equation has no solutions in natural numbers. Of course, there may be a much simpler way to prove this special case of the Catalan problem.

- Mar 2nd 2009, 02:57 PM #5

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- Mar 8th 2009, 04:43 AM #6

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## x^3 = y^2 + 2

Here's the correct question - sorry:

with .

I reckon there's only 1 sol'n, i.e. x = 3 and y = 5

However, despite all my attempts to prove this, nothing seems to work. Any thoughts? I've tried my usual bag o' tricks already...

- Mar 8th 2009, 02:45 PM #7

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I think I read in a book a long time ago that Euler spend seven years proving this result with elementary methods. However, if you know higher arithmetic, in particular then you can prove this result more easily. The steps are here.

- Mar 9th 2009, 02:17 AM #8

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- Mar 10th 2009, 05:05 AM #9
According to…

http://mathworld.wolfram.com/NaturalNumber.html

… it is controversial if 0 is included or not in the ‘natural numbers’. If we accept that 0 is a 'natural number', then is solution of the equation …

Regards

- Mar 10th 2009, 09:27 PM #10

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