T. Trudgian

activity

20152019

collaborators

9 papers

math.NT2019

Explicit upper bounds on the least primitive root

Kevin J. McGown, Tim Trudgian

We give a method for producing explicit bounds on $g (p)$ , the least primitive root modulo $p$ . Using our method we show that $g (p) < 2 r 2^{r ω (p - 1)} p^{\frac{1}{4} + \frac{1}{4 r}}$ fo…

math.NT2019

On integers $n$ for which $σ (2 n + 1) \geq σ (2 n)$

Mits Kobayashi, Tim Trudgian

We show that the natural density of positive integers $n$ for which $σ (2 n + 1) \geq σ (2 n)$ is between $0.053$ and $0.055$ .

math.NT2017

Uchiyama's conjecture on sums of squares

Tim Trudgian

Uchiyama showed that every interval $(n, n + c n^{1/4})$ contains an integer that is the sum of two squares, where $c = 2^{3/2}$ . He also conjectured a minimal value of $c$ such tha…

math.NT2017

Lehmer numbers and primitive roots modulo a prime

Stephen D. Cohen, Tim Trudgian

A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p - 1$ whose inverse $a ˉ$ within the same range has opposite parity. Lehmer numbers that are also primit…

math.NT2017

Quadratic residues that are not primitive roots

Tamiru Jarso, Tim Trudgian

We prove that any prime $p$ satisfying $ϕ (p - 1) \leq (p - 1) /4$ contains two consecutive quadratic non-residues modulo $p$ neither of which is a primitive root modulo $p$ .

math.NT2017

Square-full primitive roots

Marc Munsch, Tim Trudgian

We use character sum estimates to give a bound on the least square-full primitive root modulo a prime. Specifically, we show that there is a square-full primitive root mod $p$ less…

Explicit upper bounds on the least primitive root

On integers n for which σ(2n+1)≥σ(2n)

Uchiyama's conjecture on sums of squares

Lehmer numbers and primitive roots modulo a prime

Quadratic residues that are not primitive roots

Square-full primitive roots

On integers $n$ for which $σ (2 n + 1) \geq σ (2 n)$