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 February 2nd, 2010, 03:15 PM #1 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 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 ) 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: Integer-valued polynomials These are the Kempner numbers, Sloane's A002034. Your suspicion is right regarding primes. Tags integervalued, polynomials Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post 03sqq Applied Math 1 January 31st, 2013 05:12 AM asen_ttil Real Analysis 1 June 5th, 2012 10:32 AM asen_ttil Calculus 2 May 29th, 2012 08:25 PM playthious Calculus 3 December 16th, 2010 11:41 PM dwnielsen Real Analysis 0 December 19th, 2009 09:41 AM

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