About the diophantine equation z²= 32y² – 16

Volume 4, Issue 5, October 2019     |     PP. 126-139      |     PDF (312 K)    |     Pub. Date: November 4, 2019
DOI:    203 Downloads     4443 Views  

Author(s)

Serge PERRINE, CentraleSupelec Campus de Metz 2 rue Edouard Belin, 57070 Metz, France

Abstract
A Pell Fermat equation and its two classes of solutions are discussed. We give a formula for the pairs of positive solutions, written with the Pell numbers, and some new identities involving these numbers. We build an invariant modulo 4 for each class of solutions.

Keywords
Pell numbers, Pell-Lucas numbers, Markoff equation

Cite this paper
Serge PERRINE, About the diophantine equation z²= 32y² – 16 , SCIREA Journal of Mathematics. Volume 4, Issue 5, October 2019 | PP. 126-139.

References

[ 1 ] Andreescu, T. Andrica, D., Quadratic diophantine equations, Springer Verlag, New York, 2015.
[ 2 ] Aigner, M. Markov’s theorem and 100 years of the uniqueness conjecture, Verlag, Cham Heidelberg New York, Dordrecht, London, 2013.
[ 3 ] Emerson, E. Recurrent sequences in the equation 𝐷𝑄² = 𝑅² + 𝑁, Fibonacci Quarterly, 7, 1969, 233-242.
[ 4 ] Halter Koch, F. Quadratic irrationals - An introduction to classical number theory, CRC Press, New York, 2013.
[ 5 ] Koshy, T., Pell and Pell Lucas numbers with applications, Springer Verlag New York, 2014.
[ 6 ] LeVeque, W. J. Topics in number theory, vol. 1 and 2, Dover, New York, 2002.
[ 7 ] Matthews, K., Quadratic diophantine equations BCMATH programs, Solving 𝑥² − 𝑑𝑦² = 𝑛, 𝑑 > 0, 𝑛 non-zero: for fundamental solutions, by the Lagrange–Mollin-Matthews method http://www.numbertheory.org/php/main_pell_ html 2015.
[ 8 ] Perrine, S., Some properties of the equation 𝑥² = 5𝑦² − 4, The Fibonacci Quarterly, 54 (2), 2016, 172–178.
[ 9 ] Robertson, J. P., Characterization of fundamental solutions to generalized Pell equations, 2014, http://www.jpr2718.org/ .
[ 10 ] Sloane, N. J. A., The On-line Encyclopedia of Integer Sequences, http://oeis.org .