Tribonacci numbers and primes of the form p = x2 + 11y2
-
Tim Evink
and
Paul Alexander Helminck
Abstract
In this paper we show that for any prime number p not equal to 11 or 19, the Tribonacci number Tp−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 Tp−1. We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight 2 and level 11.
(Communicated by Filippo Nuccio)
Acknowledgement
The authors would like to thank Associate Professor Burkard Polster (from “Mathologer”) for bringing these numbers under their attention through his video on YouTube on Tribonacci numbers. The authors would also like to thank Prof. Jaap Top for pointing out the connection with modular forms and the unique newform of weight 2 and level 11 and the referees for their comments and remarks, in particular for suggesting Lemmas 3.1 and 3.2.
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© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- The life jubilee of Prof. RNDr. Sylvia Pulmannová, DrSc.
- Perfect 1-factorizations
- A topological duality for strong Boolean posets
- On the Diophantine equations x2 + 2α 3β 19γ = yn and x2 + 2α 3β 13γ = yn
- Tribonacci numbers and primes of the form p = x2 + 11y2
- Basic semirings
- A conjecture for varieties of completely regular semigroups
- Uniqueness of meromorphic function with its shift operator under the purview of two or three shared sets
- Differential subordination results for Mittag-Leffler type functions with bounded turning property
- Mittag-Leffler stability for non-instantaneous impulsive Caputo fractional differential equations with delays
- Asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2
- On the polynomial entropy for morse gradient systems
- Quantitative approximation by Stancu-Durrmeyer-Choquet-Šipoš operators
- A note on non-linear ∗-Jordan derivations on ∗-algebras
- Disjoint hypercyclic weighted translations on locally compact hausdorff spaces
- Some new results on real hypersurfaces with generalized Tanaka-Webster connection
- Relative topological properties of hyperspaces
- Cohomology of torus manifold bundles
- The Menger and projective Menger properties of function spaces with the set-open topology
- Asymptotic behavior of the record values in a stationary Gaussian sequence, with applications
Articles in the same Issue
- Regular papers
- The life jubilee of Prof. RNDr. Sylvia Pulmannová, DrSc.
- Perfect 1-factorizations
- A topological duality for strong Boolean posets
- On the Diophantine equations x2 + 2α 3β 19γ = yn and x2 + 2α 3β 13γ = yn
- Tribonacci numbers and primes of the form p = x2 + 11y2
- Basic semirings
- A conjecture for varieties of completely regular semigroups
- Uniqueness of meromorphic function with its shift operator under the purview of two or three shared sets
- Differential subordination results for Mittag-Leffler type functions with bounded turning property
- Mittag-Leffler stability for non-instantaneous impulsive Caputo fractional differential equations with delays
- Asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2
- On the polynomial entropy for morse gradient systems
- Quantitative approximation by Stancu-Durrmeyer-Choquet-Šipoš operators
- A note on non-linear ∗-Jordan derivations on ∗-algebras
- Disjoint hypercyclic weighted translations on locally compact hausdorff spaces
- Some new results on real hypersurfaces with generalized Tanaka-Webster connection
- Relative topological properties of hyperspaces
- Cohomology of torus manifold bundles
- The Menger and projective Menger properties of function spaces with the set-open topology
- Asymptotic behavior of the record values in a stationary Gaussian sequence, with applications
