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February 2nd, 2010, 03:15 PM  #1 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Integervalued polynomials
Let be a nonconstant 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 ) 
February 3rd, 2010, 06:07 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Integervalued polynomials
These are the Kempner numbers, Sloane's A002034. Your suspicion is right regarding primes.


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