The following question already on math.stackexchange asks, given integers 0 < a < b < c with gcd(a,b,c)=1, how to find three integers x,y,z for which ax+by+cz=1. I have almost the same question but I’d like to optimize x, y and z in some sense (e.g., minimal L1, L2 or L-infinity norm, I’m not picky).

something similar to the Bézout’s identity, but with three integers.

Here are some ways I’ve investigated this:

- The extended Euclid’s algorithm optimizes the solution when there are just two integers, but I don’t see a generalization of Euclid’s algorithm to multiple inputs other than picking two and repeating, as suggested in the answer to the question above.
- The Wikipedia page for Bezout’s identity mentions that the identity exists for three or more inputs, but so far I have not found anything about optimizing.
- This seems like a lattice problem but I’m not sure what the lattice would be since the solutions of Bezout’s identity do not form a group.