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

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

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

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

It would be very interesting to find a way to classify all of the exceptional pairs [TEX](n_0,n_1)[/TEX]. In particular it may give us all divisors [TEX]kb^m + 1[/TEX] of numbers [TEX]b^{8m}+1[/TEX].