We know that there is no polynomial $P(x)＝a_{n}x^n+a_{n-1}x^{n-1}+...+a_{0}$with integer coefficients and of degree at least 1 such that $P(0),P(1)...$ are all prime numbers.A simple proof can be:Assume the contrary and let $P(0)=p$, $p$ is prime.Then $P(kp)=p$ since $P(kp)$ is divisible by p.Hence $P(x)$ takes the same value infinitely many times,a contradictory.Then I want to know,if cancel the condition that $P(0)$ is prime,is the conclusion still right?By intuition I still trust there is no such polynomial,but I failed to prove.
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Any factor of $a_0$ would do instead of $p$. – Peter Feb 01 '25 at 08:59
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1@Peter: That leaves the possibilities $a_0 \in {0, \pm 1}$ though, right? – Torsten Schoeneberg Feb 01 '25 at 09:10
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2@TorstenSchoeneberg $0$ is not an issue - every number is a factor. $1$ is an issue though. – Peter Feb 01 '25 at 09:24
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If $P(x)$ were such a polynomial, then $Q(x) := P(x+1)$ has integer coefficients and $Q(0), Q(1), ...$ would be prime. So: no.
Torsten Schoeneberg
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1Please strive not to post more (dupe) answers to PSQs & dupes of FAQs. This is enforced site policy, see here. – Bill Dubuque Feb 01 '25 at 18:38
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@BillDubuque Fair enough, I was not aware of those duplicates but should have looked for them. Arguably, at least the argument here is rather different from the (objectively superior) general arguments in some of the duplicate answers. – Torsten Schoeneberg Feb 01 '25 at 19:02
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Such shift (or linear) reduction is already used in the linked dupes. And of course this specific example already occurs in prior answers too (but I see no need to search for them since the linked answers are already simple enough. The site is 15 years old so generally common exercises are dupes many times over. Please keep that in mind before answering. – Bill Dubuque Feb 01 '25 at 19:11
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I feel sorry about this.But I failed to find similar questions when editting this.It did terrify me as it's actually a simply extension.And I think it's common for new users to ask repeated questions inevitably.But,sorry for this again. – Sky Stream Feb 05 '25 at 14:44