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Abstract

The Pell sequence (Pn)n=0 is given by the recurrence Pn = 2Pn −1 + Pn −2 with initial condition P 0 = 0, P 1 = 1 and its associated Pell-Lucas sequence (Qn)n=0 is given by the same recurrence relation but with initial condition Q 0 = 2, Q 1 = 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

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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.

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Kiss, P. , Pure powers and power classes in the recurrence sequence, Math. Slovaca , 44 (1994), no. 5, 525–529. MR 96c :11017 Kiss P. Pure powers and power classes in

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-Aho , T. and Prévost , M. , Irrationality measures for the series of reciprocals from recurrence sequences , J. Number Theory 96 ( 2002 ), 275 – 292 . [13

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binary recurrence sequence , Ann. Math. Qué. , 38 ( 2 ) ( 2014 ), 169 – 188 . [17] H erschfeld , A. , The equation 2 x − 3 y = d, Bull. Amer. Math. Soc. , 41

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