Finding special factors of values of polynomials at integer points

Badziahin, D.

doi:10.1142/s1793042117500129

Finding special factors of values of polynomials at integer points

Badziahin, D.

Authors

D. Badziahin

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

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Copyright Statement
Electronic version of an article published as International Journal of Number Theory,February 2017, Vol. 13, No. 01, pp. 209-228, 10.1142/S1793042117500129 (DOI) © copyright World Scientific Publishing Company http://www.worldscientific.com/worldscinet/ijnt