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In 1995, Wiles proved that the Fermat curve does not have a solution in the positive integers when n is at least 3. However, when p is a sufficiently large prime with respect to the given Fermat curve, we know that the Fermat curve has a solution modulo p. The proof of this fact can be obtained by Schur’s theorem, which is one of the earliest results of Ramsey theory. Schur’s theorem states that whenever the positive integers are finitely coloured, there is always a monochromatic solution to the linear equation x+y=z. After giving a proof of Schur’s theorem via the finite Ramsey theorem and some related results from Ramsey theory, we consider analog equations of Fermat curves, and we show that those Diophantine equations always admit a non-trivial solution modulo p when the prime p is large enough.