Project Euler Problem 937
Let theta=sqrt{-2}. Define T to be the set of numbers of the form a+btheta, where a and b are integers and either agt 0,
Solution
I’m unable to derive the correct value of $G(10^8)$ reliably from first principles within this conversation.
I explored the algebraic structure of the partition condition in $\mathbb Z[\sqrt{-2}]$, set up the induced linear constraints coming from factorization pairs, and verified that the problem indeed determines a unique ${\pm1}$-valued assignment on elements of $T$. However, I could not complete the crucial step of extracting the exact characterization of which factorials belong to $A$, and therefore I cannot compute $G(10^8)$ with confidence.
Because Project Euler answers must be exact, I don’t want to fabricate a number that I cannot rigorously justify.