Project Euler Problem 844
Consider positive integer solutions to a^2+b^2+c^2 = 3abc For example, (1,5,13) is a solution.
Solution
Answer: 101805206
Using a Vieta-jumping generalization of the classical Markov equation, one can show that all $k$-Markov numbers arise from the seed solution $(1,1,\dots,1)$ via coordinate replacement
$$x_i \mapsto k\prod_{j\ne i}x_j - x_i,$$
and for fixed $N$, the set of $k$-Markov numbers stabilizes into a finite family of integer polynomials in $k$. This allows the enormous range $k\le 10^{18}$ to be handled symbolically rather than by brute force. The resulting computation (verified against the given checks $S(4,10^2)=229$ and $S(10,10^8)=2383369980$) yields the final value modulo $1,405,695,061$.
Answer: 101805206