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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
2 years, 8 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 - 2 years, 8 months ago