arXiv subjects

Gustav Kjærbye Bagger

Publications and source records attributed to Gustav Kjærbye Bagger.

5 recordsLinked to original sources

arXiv2026

Primitive normal progressions in finite fields

Consider an arithmetic progression inside the finite field $\mathbb{F}_{q^n}$. We study the existence of progressions whose terms are all primitive and at least one of them is normal. Using a combination of sieve estimates, ramified bounds, and computational arguments, we improve significantly upon previous literature for progressions of length $m=4$ and obtain partial results whenever $5\leq m\leq 10$. Finally, we provide a general bound which outperforms previous results in the range $m\geq 53$.

math.NT↗

arXiv2026

The determination of norm-Euclidean cyclic cubic fields

It is known on the Generalised Riemann Hypothesis that there are precisely $13$ cyclic cubic fields that are norm-Euclidean. Unconditionally, there is a gap between analytic estimates which hold for all sufficiently large conductors and computational techniques. In this paper, we establish new results concerning explicit bounds for cubic non-residues and refine previous computational techniques, enabling us to completely characterise all norm-Euclidean cyclic cubic fields.

math.NT↗

arXiv2026

A note on least primitive roots

The least primitive root $g(p)$ modulo a prime $p$ is conjectured by Grosswald to satisfy $g(p)<\sqrt{p}-2$ for any $p>409$. We make progress towards the conjecture by proving $g(p)<\sqrt{p}-2$ when $p > 1.77 \times 10^{54}$.

math.NT↗

arXiv2025

The modified prime sieve for primitive elements in finite fields

Let $r \geq 2$ be an integer, $q$ a prime power and $\mathbb{F}_{q}$ the finite field with $q$ elements. Consider the problem of showing existence of primitive elements in a subset $\mathcal{A} \subseteq \mathbb{F}_{q^r}$. We prove a sieve criterion for existence of such elements, dependent only on an estimate for the character sum $\sum_{γ\in \mathcal{A}}χ(γ)$. The flexibility and direct applicability of our criterion should be of considerable interest for problems in this field. We demonstrate the utility of our result by tackling a problem of Fernandes and Reis (2021) with $\mathcal{A}$ avoiding affine hyperplanes, obtaining significant improvements over previous knowledge.

math.NT↗

arXiv2024

Hybrid bounds for prime divisors

Let $x$ and $n$ be positive integers. We prove a non-trivial lower bound for $x$, dependant only on $ω_n$, the number of distinct prime factors of $x^n-1$. By considering the divisibility of $φ\mid x^n-1$ for $φ\mid n$, we obtain a further refinement. This bound has applications for existence problems relating to primitive elements in finite fields.

math.NT↗