Summary
This paper looks at special sets of numbers inside certain complex number systems called imaginary quadratic fields, focusing on how these sets avoid covering all possible remainders when divided by prime elements. The authors investigate the size and spacing of these sets, which relate to understanding how closely together prime numbers can appear in these complex systems. Their computations find new records for the smallest distances between primes in these fields and show specific cases where a famous prime distribution conjecture does not hold. This helps both explore prime number behavior beyond just regular integers and test long-standing mathematical predictions.
What this means in practice
- •For cryptography engineers: Refine prime selection algorithms by understanding prime distribution patterns in complex number rings related to cryptographic structures.
- •For algorithm developers: Optimize number-theoretic computations involving prime gaps by leveraging newly discovered spacings in imaginary quadratic integer rings.
Abstract
Let $\mathcal O_K$ be the ring of integers of a number field $K$. A finite subset $\mathcal H=\{η_1,\ldots,η_k\}\subset\mathcal O_K$ is said to be \emph{admissible} if for every prime ideal $\mathfrak p$ in $\mathcal O_K$, the reductions of the $η_i$ modulo $\mathfrak p$ do not represent every residue class. The diameter of a finite subset $S$ of $\mathcal O_K$ is defined to be \[\mathrm{diam}(S):= \max_{η_i,η_j\in S:η_i\neqη_j}|\mathcal N(η_i-η_j)|\] where $\mathcal N$ denotes the norm down to the rational field. This paper computationally investigates admissible subsets of rings of integers in imaginary quadratic fields of class number 1 (such as the Gaussian integers or the Eisenstein integers) that are large relative to their diameter. The motivation is twofold. Firstly, there is the improvement of results analogous to the question of an upper bound for the least gap between large rational primes, which has been one of the great number theoretic successes of the twenty-first century. A second motivation is that there is the opportunity to disprove a variation of the second Hardy-Littlewood conjecture. The calculations in the paper provide new records on the smallest distance between primes in the fields in question, and explicit counter-examples to a generalisation of the second Hardy-Littlewood conjecture in all bar one of the fields.