# A question on Bernoulli polynomials

Denote by $$B_r$$ the $$r$$-th Bernoulli polynomial. Are there any positive integers $$r, x$$ such that. $$B_r(x)$$ divides $$B_r(x+1)$$ or vice versa ?

We know that $$B_n(x)- B_n\in \mathbb Z$$, so for even $$n$$ and integer $$x$$ number $$B_n(x)$$ is not an integer.
Let $$n$$ be odd. In this case $$B_n=0$$ and $$B_n(x)=n(1^{n-1}+\cdots+ (x-1)^{n-1}).$$ If $$B_n(x)\mid B_{n+1}(x)$$ then from formula $$B_n(x+1)-B_n(x)=nx^{n-1}$$ follows that $$B_n(x)\mid nx^{n-1}$$ i.e. $$1^{n-1}+\cdots+ (x-1)^{n-1}\mid x^{n-1}.$$ After this step you'll come to the question similar to Erdős–Moser equation.