• Publications
  • Influence
The Burgess inequality and the least kth power non-residue
The Burgess inequality is the best upper bound we have for incomplete character sums of Dirichlet characters. In 2006, Booker gave an explicit estimate for quadratic Dirichlet characters which heExpand
  • 18
  • 3
  • PDF
The least k-th power non-residue
Abstract Let p be a prime number and let k ≥ 2 be a divisor of p − 1 . Norton proved that the least k-th power non-residue mod p is at most 3.9 p 1 / 4 log ⁡ p unless k = 2 and p ≡ 3 ( mod 4 ) , inExpand
  • 10
  • 2
The least kth power non-residue I
Let p be a prime number and let k ≥ 2 be an integer such that k divides p − 1. Norton proved that the least k-th power non-residue modp is at most 3.9p log p unless k = 2 and p ≡ 3 (mod 4), in whichExpand
  • 1
  • 1
  • PDF
The Primes that Euclid Forgot
TLDR
We give a completely elementary proof of Booker's result, suitable for presentation in a first course in number theory. Expand
  • 9
  • 1
  • PDF
On the sum of $k$-th powers in terms of earlier sums
For $k$ a positive integer let $S_k(n) = 1^k + 2^k + \cdots + n^k$, i.e., $S_k(n)$ is the sum of the first $k$-th powers. Faulhaber conjectured (later proved by Jacobi) that for $k$ odd, $S_k(n)$Expand
  • 1
  • 1
  • PDF
The multidimensional Frobenius problem
Consider the problem of determining maximal vectors g such that the Diophantine system Mx = g has no solution. We provide a variety of results to this end: conditions for the existence of g,Expand
  • 5
  • PDF
The least inert prime in a real quadratic field
TLDR
We prove that for any positive fundamental discriminant D > 1596, there is always at least one prime p ≤ D0.45 such that the Kronecker symbol (D/p) = −1. Expand
  • 5
  • PDF
Numerically explicit estimates for character sums
Character sums make their appearance in many number theory problems: showing that there are infinitely many primes in any coprime arithmetic progression, estimating the least quadratic non-residue,Expand
  • 7
  • PDF
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 eExpand
  • 8
  • PDF
ON THE MAXIMUM NUMBER OF CONSECUTIVE INTEGERS ON WHICH A CHARACTER IS CONSTANT
Let χ be a non-principal Dirichlet character to the prime modulus p. In 1963, Burgess showed that the maximum number of consecutive integers H for which χ remains constant is O ( p1/4 log p ) . ThisExpand
  • 4
  • PDF
...
1
2
3
4
...