The Chabauty–Kim method and its refined variant by Betts and Dogra aim to cut out the S-integral points \(X(\mathbb {Z}_S)\) on a curve inside the p-adic points \(X(\mathbb {Z}_p)\) by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line when S contains two primes. We describe an algorithm for computing refined Chabauty–Kim loci and verify Kim’s Conjecture over \(\mathbb {Z}[1/6]\) for all choices of auxiliary prime \(p < 10{,}000\) .
