Exponential Diophantine Equation Troubles

Hello, fellow Brillianters, how are you all doing?

Recently I came across this question here, and whilst trying to come up with a proof for my solution, I stumbled upon this equation here:

\(2*5^{a} - 1 = b^{2}\), where both a and b are non-negative integers.

I wanted to prove that there are only three pairs of solutions \((a,b)\) for this question; namely, \((0, 1)\), \((1, 3)\) and \((2, 7)\).

My first impulse was to try to prove that if any odd integer \(b\), \(b > 7\), is not a solution, then \(b + 4\) cannot be a solution as well. I thought that it was sufficient until Calvin Lin came along and showed me that I only proved that \(b\) and \(b + 4\) cannot be solutions at the same time. Worst part is that he has no idea either of how to prove this.

So here I am, my friends; do you know a way to prove my statement right (or wrong)? I'd appreciate any form of help you can provide me. Thanks!

Note by Alexandre Miquilino
3 years, 2 months ago

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This is a Ramanujan-Nagell type equation.

According to Wikipedia, a result of Siegel implies that the number of solutions in each case is finite, but not much further is known.

I believe It is unlikely that there is a simply proof of this statement.

Calvin Lin Staff - 3 years, 2 months ago

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