# 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
5 years, 4 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.

Staff - 5 years, 4 months ago