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:

25a1=b22*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)(a,b) for this question; namely, (0,1)(0, 1), (1,3)(1, 3) and (2,7)(2, 7).

My first impulse was to try to prove that if any odd integer bb, b>7b > 7, is not a solution, then b+4b + 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 bb and b+4b + 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
6 years, 5 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 - 6 years, 5 months ago

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