The least primitive root modulo $p^{2}$
We provide an explicit estimate on the least primitive root mod $p^{2}$. We show, in particular, that every prime $p$ has a primitive root mod $p^{2}$ that is less than $p^{0.99}$.
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arXiv subjects
Tim Trudgian
Publications and source records attributed to Tim Trudgian.
Explicit upper bounds on the least primitive root
We give a method for producing explicit bounds on $g(p)$, the least primitive root modulo $p$. Using our method we show that $g(p)<2r\,2^{rω(p-1)}\,p^{\frac{1}{4}+\frac{1}{4r}}$ for $p>10^{56}$ where $r\geq 2$ is an integer parameter. This result beats existing bounds that rely on explicit versions of the Burgess inequality. Our main result allows one to derive bounds of differing shapes for various ranges of $p$. For example, our method also allows us to show that $g(p)<p^{5/8}$ for all $p\geq 10^{22}$ and $g(p)<p^{1/2}$ for $p\geq 10^{56}$.
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On integers $n$ for which $σ(2n+1)\geq σ(2n)$
We show that the natural density of positive integers $n$ for which $σ(2n+1)\geq σ(2n)$ is between $0.053$ and $0.055$.
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An elementary bound on Siegel zeroes
We consider Dirichlet $L$-functions $L(s, χ)$ where $χ$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\geq 3$ the function $L(s, χ)$ has at most one real zero $β$ with $1- 1.011/\log q < β< 1$.
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Existence results for primitive elements in cubic and quartic extensions of a finite field
With $\Fq$ the finite field of $q$ elements, we investigate the following question. If $γ$ generates $\Fqn$ over $\Fq$ and $β$ is a non-zero element of $\Fqn$, is there always an $a \in \Fq$ such that $β(γ+ a)$ is a primitive element? We resolve this case when $n=3$, thereby proving a conjecture by Cohen. We also improve substantially on what is known when $n=4$.
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Linear combinations of primitive elements of a finite field
We examine linear sums of primitive roots and their inverses in finite fields. In particular, we refine a result by Li and Han, and show that every $p> 13$ has a pair of primitive roots $a$ and $b$ such that $a+ b$ and $a^{-1} + b^{-1}$ are also primitive roots mod $p$.
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Primitive values of quadratic polynomials in a finite field
We prove that for all $q>211$, there always exists a primitive root $g$ in the finite field $\mathbb{F}_{q}$ such that $Q(g)$ is also a primitive root, where $Q(x)= ax^2 + bx + c$ is a quadratic polynomial with $a, b, c\in \mathbb{F}_{q}$ such that $b^{2} - 4ac \neq 0$.
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Improved bounds on Brun's constant
Brun's constant is $B=\sum_{p \in P_{2}} p^{-1} + (p+2)^{-1}$, where the summation is over all twin primes. We improve the unconditional bounds on Brun's constant to $1.840503 < B < 2.288513$, which is about a 13\% improvement on the previous best published result.
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Fujii's development on Chebyshev's conjecture
Chebyshev presented a conjecture after observing the apparent bias towards primes congruent to $3\pmod 4$. His conjecture is equivalent to a version of the Generalised Riemann Hypothesis. Fujii strengthened this conjecture; we strengthen it still further using detailed computations of zeroes of Dirichlet $L$-functions.
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Uchiyama's conjecture on sums of squares
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 that the above statement still holds. We investigate this claim.
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Lehmer numbers and primitive roots modulo a prime
A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an endeavour to count the number of ways $1$ can be expressed as the sum of two primitive roots that are also Lehmer numbers (an extension of a question of S. Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo $p$ and prove that, for all primes $p \neq 2,3,7$, Lehmer primitive roots exist. We also make explicit the known expression for the number of Lehmer numbers modulo $p$ and improve the Wang--Wang estimate for the number of solutions to the Golomb--Lehmer primitive root problem.
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Quadratic residues that are not primitive roots
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 $p$.
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Square-full primitive roots
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 than $p^{2/3 + 3/(4 \sqrt{e})+ ε}$, and we give some conditional bounds.
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On the sum of two squares and at most two powers of 2
We demonstrate that there are infinitely many integers that cannot be expressed as the sum of two squares of integers and up to two non-negative integer powers of 2.
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On the least square-free primitive root modulo $p$
Let $g^{\square}(p)$ denote the least square-free primitive root modulo $p$. We show that $g^{\square}(p)< p^{0.96}$ for all $p$.
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Resolving Grosswald's conjecture on GRH
In this paper we examine Grosswald's conjecture on $g(p)$, the least primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald's conjecture by showing that $g(p)< \sqrt{p} - 2$ for all $p>409$. Our method also shows that under GRH we have $\hat{g}(p)< \sqrt{p}-2$ for all $p>2791$, where $\hat{g}(p)$ is the least prime primitive root modulo $p$.
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Searching for Diophantine quintuples
We consider Diophantine quintuples $\{a, b, c, d, e\}$. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most $1.18\cdot 10^{27}$ Diophantine quintuples.
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On Grosswald's conjecture on primitive roots
Grosswald's conjecture is that $g(p)$, the least primitive root modulo $p$, satisfies $g(p) \leq \sqrt{p} - 2$ for all $p>409$. We make progress towards this conjecture by proving that $g(p) \leq \sqrt{p} -2$ for all $409 3.67\times 10^{71}$.
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