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The solutions of $a^2 = b^d -3^c$ are in the form $(a, b, c, d) = ((46)27^t, (13)9^t, 6t+4, 3)$. This is done by using calculator. As per my calculator, I have checked some terms, which are satisfied the above cited equation. Now, my question is, is there any other forms of solutions? if there, how to examine?

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    @GerryMyerson!this is somewhat good.2012-10-20

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One infinite, albeit somewhat trivial, family of solutions has $d=2$. This leads to $b^2-a^2=3^c$. It is well-known, how to express a number as a difference of two squares. For any $r\lt c/2$, we get $b=(3^r+3^{c-r})/2$, $a=(3^{c-r}-3^r)/2$.

Another solution is given by $10^2=7^3-3^5$.

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    ! without taking d= some fixed number, can we get a, b, c and d values in terms of single variable, which I have shown in the post. I need in single variable with proof. I am not looking cases.2012-10-20