Tribonacci numbers and primes of the form $p=x^2+11y^2$

TL;DR

This paper establishes a deep link between primes of the form x^2+11y^2 and the divisibility of Tribonacci numbers, using advanced algebraic number theory and modular forms.

Contribution

It proves a new characterization of primes related to Tribonacci numbers via class field theory and modular forms, extending classical prime representation results.

Findings

Primes p ≠ 11, 19 satisfy p | T_{p-1} iff p = x^2 + 11y^2.

Connection established between Tribonacci numbers and Fourier coefficients of a weight 2 level 11 modular form.

Uses class field theory to analyze prime splitting in a specific cubic field.

Abstract

In this paper we show that for any prime number $p$ not equal to $11$ or $19$, the Tribonacci number $T_{p-1}$ is divisible by $p$ if and only if $p$ is of the form $x^2+11y^2$. We first use class field theory on the Galois closure of the number field corresponding to the polynomial $x^3-x^2-x-1$ to give the splitting behavior of primes in this number field. After that, we apply these results to the explicit exponential formula for $T_{p-1}$. We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight $2$ and level $11$.