On the continued fraction expansion of the unique root in F(p) of the equation x^4+x^2-Tx-1:12=0 and other related hyperquadratic expansions

TL;DR

This paper investigates the continued fraction expansions of solutions to a specific degree 4 algebraic equation over finite fields, revealing patterns depending on the prime's residue modulo 3, with detailed analysis for p=7 and p=13.

Contribution

It identifies and describes the patterns of continued fraction expansions for solutions of a hyperquadratic equation over finite fields, especially for primes p=7 and p=13.

Findings

Two distinct continued fraction patterns depending on p mod 3

Explicit description of the pattern for p=7 and p=13

Initial observations and indications for other primes

Abstract

In 1985, Robbins observed by computer the continued fraction expansion of certain algebraic power series over a finite field. Incidentally, he came across a particular equation of degree 4 in characteristic p=13. This equation has an analogue for all primes p>=5. There are two patterns for the continued fraction of the solution of this equation, according to the residue of p modulo 3. We describe this pattern in the first case, considering especially p=7 and p=13. in the second case we only give indications.