Research PapersNo Access

Finding special factors of values of polynomials at integer points

Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Road, Durham DH1 3LE, UK

E-mail Address: dzmitry.badziahin@durham.ac.uk

https://doi.org/10.1142/S1793042117500129Cited by:0 (Source: Crossref)

Abstract

We investigate the divisors $d$ of the numbers $P (n)$ for various polynomials $P \in ℤ [x]$ such that $d \equiv 1 (mod n)$ . We obtain the complete classification of such divisors for a class of polynomials, in particular for $P (x) = x^{4} + 1$ . We also construct a fast algorithm which provides all such factorizations up to a given limit for another class, for example for $P (x) = 2 x^{4} + 1$ . We use these results to find all the divisors $d = 2^{m} k + 1$ of numbers $2^{4 m} + 1$ and $2^{4 m + 1} + 1$ . For the numbers $2^{4 m} + 1$ the complete classification of such divisors is provided while for the numbers $2^{4 m + 1} + 1$ the given classification is proved to be exhaustive only for $m \leq 1000$ .

Keywords:

AMSC: 11Y05, 11D99, 11C08

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