Does anyone know how i'd solve a set of equations like this?
I'm trying to find valid values of $x_i$ and $k_i$. I've found some solutions using a program to brute force it, but I want to up the number of $x$'s, $k$'s which rules out brute force as a practical solution.
For instance, there are two $x$ values above, which translate to $2^x$ $k$ values, and $x∗2^x$ equations. I'd like to take the number of $x$ values up to 16, or 32 if possible, which results in huge numbers of $k$'s and equations.
Anyone able to help at all, even to point me in some direction?
I do know about the chinese remainder theorem, multiplicative modular inverse and the extended euclidean algorithm, among some other basic modulus math techniques, but I'm not really sure how to make any progress on this.
Edit: To clarify a bit, Ideally I'd like to find all solutions to this problem, but if there is a way to find a subset of solutions, like if the equations below could be solved that would be fine too. Or, if there is some way to find solutions numerically which is much faster than brute force permuting the $x_i$ and $k_i$ values and testing if they fit the constraints, that'd be helpful too.