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Old 2014-11-28, 14:51   #1
Drdmitry's Avatar
Nov 2011

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Default Finding all divisors kn + 1 of P(n) for various polynomials P

Let P(x)\in Z[x] be a monic polynomial with such that x^nP(-x^{-1}) = P(x). I was interested in looking at the divisors of the values P(n) of the form kn+1. It appears that, given one divisor
n_0n_1+1 | P(n_1),
there is an infinite series of the divisors of this form which is given by the equations
P(n_k) = (n_{k-1}n_k+1)(n_{k+1}n_k+1).
For the polynomial P(n) = n^4 + 1 one can classify all such series. They are "generated" by the pairs (n_0, n_1) = (0, n) for an arbitrary n\in N. This in turn implies (with some efforts to be made) that numbers b^{4m}+1 do not have (non-algebraic) divisors of the form kb^m + 1.

Concerning the polynomial P(n) = n^8 + 1 we also have series of divisors of P(n) generated by the pairs (n_0, n_1) = (0,n). Additionally the pairs (n_0, n_1) = (n^3, n^5) generate infinite series of divisors of P(n). However a basic search among small numbers shows that there are still "exceptional" pairs (n_0,n_1) which generate the divisors n_0n_1+1 | n_1^8 + 1, the smallest of them is (3,11). With some efforts one can check that all (non-algebraic) divisors kb^m+1 of numbers b^{8m}+1 must come from an "exceptional" pair (k,b^m).

I do not know how to classify the "exceptional" pairs (n_0, n_1). I conducted a search for all pairs with \max\{n_0,n_1\} \le 10^6 and additionally with n_0+n_1\le 10^7. In total there are 201 different infinite series found. Also it seems that they are more less equidistributed on a \log n_0 \times \log n_1 coordinate plane.

It would be very interesting to find a way to classify all of the exceptional pairs (n_0,n_1). In particular it may give us all divisors kb^m + 1 of numbers b^{8m}+1.
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File Type: pdf spec_fact.pdf (199.9 KB, 171 views)
File Type: txt x^8+1_chains.txt (10.3 KB, 236 views)
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