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February 2nd, 2010, 03:15 PM   #1
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Integer-valued polynomials

Let be a non-constant polynomial with integer coefficients and a leading coefficient of 1, so that

for some with Moreover, we will say that an integer divides (as usual, denoted ) if for all we have

For , let denote the smallest integer such that there exists a polynomial as described above with and is of degree

For example, since satisfies the above description with

since is always even, while is never even for all

since is always divisible by 3, and this is the smallest degree for which this is possible;

- consider

- consider

- consider

Can we find the general term for Clearly since for all I suspect that we will also find that for prime.

(Note that this problem is equivalent to finding the largest such that are linearly independent modulo )
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February 3rd, 2010, 06:07 AM   #2
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Re: Integer-valued polynomials

These are the Kempner numbers, Sloane's A002034. Your suspicion is right regarding primes.
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