Combinatorial numbers in binary recurrences

Abstract

Abstract. We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally, partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.

On special extrema of polynomials with applications to Diophantine problems

“…The latter problem attracted a lot of attention already, and our Theorem 3.5 generalizes and/or extends several results from the literature, e.g. those of Pethő [31] and Shorey and Stewart [39] about perfect powers in non-degenerate binary recurrence sequences, Nemes and Pethő [29] concerning values of general polynomials in certain binary recurrence sequences and of Kovács [24] on values of certain combinatorial polynomials in concrete, important binary recurrence sequences. We give more details before the formulation of Theorem 3.5.…”

“…As we see, Theorem 3.5 can be considered to be a generalization of the above mentioned results from [24,31,39,40,44]. Further, it provides an effective version of the main result from [29].…”

“…Finally, we mention that there are many results in the literature where values of special polynomials in specific binary recurrence sequences are studied (see e.g. the paper Kovács [24] and the references given there).…”

On Diophantine equations involving Lucas sequences

“…This was proved by Ming [14]. Many other similar equations have been considered during the years, we cite [5] and references therein for the state of the art of these kind of problems.…”

Lucas numbers of the form (2t/k)

“…Here P m is a term of the Pell sequence. Later, Szalay [15] treated the equations F m = n 4 , L m = n 4 , and Kovács [6] solved the analogous equation P m = n 4 . The more complicated problem L m = n 5 was handled by Tengely [17].…”