Anton Huynh

2022-11-23

Polynomial problem involving divisibility, prime numbers, monotony
Let f be a polynomial function, with integer coefficients, strictly increasing on $\mathbb{N}$ such that $f\left(0\right)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio $r>0$ such that the value of function f in every term of the progression is a prime number. I believe that the solution includes a reductio ad absurdum, but I don't know how to solve it.

cenjene9gw

Expert

Explanation:
If progression is $a+nb$, and $f\left(a\right)=p$, then $f\left(a+pnb\right)\equiv 0\phantom{\rule{0.444em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}p\right)$ and absolute value is more then p for large n.

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