I need to find the number of pairs of integers (a,b) that satisfy the diophantine equation $a^3-117b^3 = 5$.
I know that I need to use modulo arithmetic somehow, I just don't know exactly how. I think it has something to do with $mod$ $9$, but to be honest I don't really know. Anything that could help me would be much appreciated.
For any n, if (a, b) is a solution of your equation, then (a mod n, b mod n) is a solution of your equation mod n. Your equation mod 9 is $a^3 = 5$ (because 117 = 9 *13). You can compute a^3 mod 9 for a = 0,1,...,8.
You get : \begin{align} &a&a^3\\ &0&0\\ &1&1\\ &2&8\\ &3&0\\ &4&1\\ &5&8\\ &6&0\\ &7&1\\ &8&8\\ \end{align} As you can see, this equation has no solutions which means your original equation doesn't have any solutions.