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The Bernoulli polynomials B n (x) are defined by the generating series The Bernoulli polynomials B n (x) are related to the sums of nth powers of natural numbers as follows. For any n ≥ 1, the sum 1 n + 2 n + · · · + k n is a polynomial function S n (k) of k andn + 1 .In this paper, for nonzero rational numbers a, b and rational polynomials C(y), we study the Diophantine equation aB m (x) = bB n (y) + C(y)

Let a, b be nonzero rational numbers and C(y) a polynomial with rational coefficients. We study the Diophantine equations and aBm(x)=bfn(y)+C(y) afm(x)=bBn(y)-I-C(y) with m ~> n > deg C + 2 for solutions in integers x, y. Here fn (x) = x (x + 1)-.. (x + n-1) and the Bernoulli polynomials Bn (x) are defined by the generating series tetX ~ tn e'-1 = ~ B,(x)~.. n=0 Then, Bn(x) = ~i~=o (n)Bn_ixi where Br = Br(O) is the rth Bernoulli number. In fact, Br are rational numbers defined recursively by Bo = 1 and n-1 ~i=0 ('~)Bi =0 for all n ~> 2. The odd Bernoulli number