Finding special factors of values of polynomials at integer points

Badziahin, D.

Abstract

We investigate the divisors dd of the numbers P(n)P(n) for various polynomials P∈Z[x]P∈ℤ[x] such that d≡1(modn)d≡1(modn). We obtain the complete classification of such divisors for a class of polynomials, in particular for P(x)=x4+1P(x)=x4+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)=2x4+1P(x)=2x4+1. We use these results to find all the divisors d=2mk+1d=2mk+1 of numbers 24m+124m+1 and 24m+1+124m+1+1. For the numbers 24m+124m+1 the complete classification of such divisors is provided while for the numbers 24m+1+124m+1+1 the given classification is proved to be exhaustive only for m≤1000m≤1000.

Citation

Badziahin, D. (2017). Finding special factors of values of polynomials at integer points. International Journal of Number Theory, 13(01), 209-228. https://doi.org/10.1142/s1793042117500129

• Journal Article Type: Article
• Acceptance Date: Jan 29, 2016
• Online Publication Date: May 6, 2016
• Publication Date: Feb 1, 2017
• Deposit Date: Jan 29, 2016
• Publicly Available Date: May 6, 2017
• Journal: International Journal of Number Theory
• Print ISSN: 1793-0421
• Electronic ISSN: 1793-7310
• Publisher: World Scientific Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 13
• Issue: 01
• Pages: 209-228
• DOI: https://doi.org/10.1142/s1793042117500129
• Public URL: https://durham-repository.worktribe.com/output/1390528