Finding special factors of values of polynomials at integer points

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe investigate the divisors d of the numbers P (n) for various polynomials P ∈ Z[x] such that d ≡ 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) = 2x 4 + 1. We use these results to find all the divisors d = 2 m k + 1 of numbers 2 4m + 1 and 2 4m+1 + 1. For the numbers 2 4m + 1 the complete classification of such divisors is provided while for the numbers 2 4m+1 + 1 the given classification is proved to be exhaustive only for m 1000.

On the irrationality measure of the Thue–Morse constant

“…This fact can be checked by substituting the functional equation (28) into the condition of Theorem L. The reader can also compare with [5,Lemma 3].…”

“…Bugeaud, Han, Wen and Yao [11] provided the following result that links the continued fraction of f (z) and the irrationality exponents of values f (b), b ∈ N. In fact, they formulated it in terms of Hankel determinants. The present reformulation can be found in [5]:…”

“…We will consider this result in more details in the next subsection. Later, Badziahin [5] provided a continued fraction expansion for the functions of the form…”

Continuant Diophantine equations

“…Proof. Firstly note that coprimeness conditions and recurrent formula (1) imply that for any m, k ∈ Z 0 such that 0…”

“…To the best of authors knowledge, equations of the form (3) are mostly uncovered in the literature. We can only refer to a paper [1] where the case n = 2 and some particular types of the polynomials P (x) were considered. On the other hand, continuants play an important role in the solutions of some Diophantine equations.…”