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

Abstract

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.