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.

No abstract

No abstract

For the hyperelliptic curve C p with equationwith p a prime number, we discuss bounds for the rank of its Jacobian over Q, find many cases having 2-torsion in the associated Shafarevich-Tate group, and we present some results on rational points of C p . c

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form p = 8k + 1, receives attention.