From where this theorem about integer valued polynomials is coming from?

Question

In his article "Polynomials with Integer values" (Resonance), Sury proved an interesting lemma:

If $P$ is a non constant, integral valued, polynomial, then the number of prime divisors of his values $\{P(m)\}_{m\in \mathbb Z}\ $ is infinite, i.e. not all the terms of the sequence $P(0),\ P(1),\ P(2)\ldots $ can be build from finitely many primes.

Does anyone know from where this theorem is taken, or is it a discovery of Sury?

• Servaes: The set of numbers that are divisible by a given finite set of primes thins out rather quickly; it forces the values of $P$ to grow exponentially. The idea is simple but the details are a bit cumbersome.
• MikeTeX: @Servaes. Do you mean you have a proof different from that of Sury?
• Servaes: I don't know what Sury's proof is, so I don't know whether my proof is different.
• MikeTeX: So, just give the idea of your proof in an answer! (in a form a bit more elaborated than you comment)

Answer 1

As per my information, Schur proved this result for the first time. I also gave an alternative proof to this result which you may find here

The reference of Schur's work can be found in the reference section of the paper.

Thanks.

• MikeTeX: The link to your paper don't work.
• देवेन्द्र prasad: sorry,check now.
• MikeTeX: Thank you. You may be interested in my own article, which is quite elementary too: arxiv.org/pdf/2112.14711.pdf
• देवेन्द्र prasad: I read the article. You could use the standard notation Int(R) for the ring of integrally valued polynomial. Thanks.