Share This Article:

Squares from D(–4) and D(20) Triples

DOI: 10.4236/apm.2011.15052    2,700 Downloads   5,917 Views
Author(s)    Leave a comment
We study the eight infinite sequences of triples of natural numbers A=(F2n+1,4F2n+3,F2n+7), B=(F2n+1,4F2n+5,F2n+7), C=(F2n+1,5F2n+1,F2n+3), D=(F2n+3,4F2n+1,F2n+3) and A=(L2n+1,4L2n+3,L2n+7), B=(L2n+1,4L2n+5,L2n+7), C=(L2n+1,5L2n+1,L2n+3), D=(L2n+3,4L2n+1,L2n+3. The sequences A,B,C and D are built from the Fibonacci numbers Fn while the sequences A, B, C and D from the Lucas numbers Ln. Each triple in the sequences A,B,C and D has the property D(-4) (i. e., adding -4 to the product of any two different components of them is a square). Similarly, each triple in the sequences A, B, C and D has the property D(20). We show some interesting properties of these sequences that give various methods how to get squares from them.

KEYWORDS

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Z. Čerin, "Squares from D(–4) and D(20) Triples," Advances in Pure Mathematics, Vol. 1 No. 5, 2011, pp. 286-294. doi: 10.4236/apm.2011.15052.

  E. Brown, “Sets in Which Is Always a Square,” Mathematics of Computation, Vol. 45, No. 172, 1985, pp. 613-620.  N. Sloane, On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/ njas/sequences/.  L. Euler, “Commentationes Arithmeticae I,” Opera Omnia, Series I, volume II, B.G. Teubner, Basel, 1915.  Z. ?erin, “On Pencils of Euler Triples I,” (in press).  Z. ?erin, “On Pencils of Euler Triples II,” (in press).  M. Radi?, “A Definition of Determinant of Rectangular Matrix,” Glasnik Mate-maticki, Vol. 1, No. 21, 1966, pp. 17-22.

comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc. This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.