That blog post is specifically about the question about whether a polynomial taking integer values on positive integers (or any sufficiently large subset thereo

That blog post is specifically about the question about whether a polynomial taking integer values on positive integers (or any sufficiently large subset thereof) necessarily is an integer-valued polynomial (the answer is yes). The OP blog post here actually links to / points out there's an Wikipedia article on integer-valued polynomials: https://en.wikipedia.org/wiki/Integer-valued_polynomial https://en.wikipedia.org/wiki/Integer-valued_polynomial Among other things, as mentioned, “every integer-valued polynomial can be written as an integer linear combination of binomial coefficients” [in exactly one way]. The conjecture in the OP post, that every polynomial everywhere divisible by k counts {something}, is intriguing, and I wouldn't be surprised if it were true.