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Let $ p$ be a prime number. For every $ n \in \mathbb N$ , let $ A_{p,n}:=\{\deg P(X) : P(X)\in \mathbb Z[X]$ is monic and $ p^n|P(m), \forall m \in \mathbb Z$ $ \}$ ; then how to show that $ A_{p,n}$ is non-empty ? If we define $ f(n,p):=\min A_{p,n}$ , then howRead more

Let $ X$ a affine algebraic variety. We have that $ X$ can have infinite isomorphisms, for example let $ X=\mathbb{A}^1$ . Let $ G$ be a divisible group. My question is about some condition in $ G$ such that I can garantize the existence of a algebraic variety $ X$ such that $ G$Read more

How to solve this question. Thanks The number 23_56 is divisible by 13. What is the missing digit?Read more