Tribonacci numbers and primes of the form $p=x²+11y²$

Year, pages: 2019, 521 - 532

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 $x²+11y²$. We first use class field theory on the Galois closure of the number field corresponding to the polynomial $x³-x²-x-1$ 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 $11$.

Evink, T., Helminck, P. 2019. Tribonacci numbers and primes of the form $p=x²+11y²$. In Mathematica Slovaca, vol. 69, no.3, pp. 521-532. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0244

Evink, T., Helminck, P. (2019). Tribonacci numbers and primes of the form $p=x²+11y²$. Mathematica Slovaca, 69(3), 521-532. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0244