APM  Vol.1 No.5 , September 2011
Squares from D(–4) and D(20) Triples
Abstract: 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.
Cite this paper: nullZ. 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.

[1]   E. Brown, “Sets in Which Is Always a Square,” Mathematics of Computation, Vol. 45, No. 172, 1985, pp. 613-620.

[2]   N. Sloane, On-Line Encyclopedia of Integer Sequences. njas/sequences/.

[3]   L. Euler, “Commentationes Arithmeticae I,” Opera Omnia, Series I, volume II, B.G. Teubner, Basel, 1915.

[4]   Z. ?erin, “On Pencils of Euler Triples I,” (in press).

[5]   Z. ?erin, “On Pencils of Euler Triples II,” (in press).

[6]   M. Radi?, “A Definition of Determinant of Rectangular Matrix,” Glasnik Mate-maticki, Vol. 1, No. 21, 1966, pp. 17-22.