A Class Of Diophantine Equations With Three Bernoulli Polynomials

Diophantine equations is one of the central problems of number theory since ancient times. For example, Fermat's last theorem, Pell equations, BSD conjecture are directly re-lated with Diophantine equations.Kulkarni and Sury first study a class of Diophantine equations with one Bernoulli poly-nomial, which is Bm(x)=g(y). They then study a class of Diophantine equations with two Bernoulli polynomials, which is Bm(x)=aBn(y)+C(y). At the same time they give a theorem that determines such Diophantine equations have only finitely many rational solutions with bounded denominators. To some extent, this theorem is a corollary of Bilu-Tichy's important theorem on Diophantine equation f(x)=g(y). Bilu and Tichy give an condition which is equivalent to that Diophantine equation f(x)=g(y) has infinitely many rational solutions with a bounded denominator, Kulkarni and Sury repeat use of this important theorem when they prove their theorem.In the first part, we will a generalization of Kulkarni-Sury's theorem which determines Diophantine equations with three Bernoulli polynomials have only finitely many rational so-lutions with bounded denominators. This paper is organized as follows, the first chapter introduces Kulkarni-Sury's theorem and the author's conclusions; the second chapter intro-duces essential knowledge to prove the theorem, including Bernoulli polynomials and their decomposition, Bilu-Tichy's important theorem; the third chapter is to prove the conclusions proposed in the first chapter.