Squares from *D*(–4) and *D*(20) Triples

Author(s)
Zvonko Čerin

ABSTRACT

We study the eight infinite sequences of triples of natural numbers*A*=(*F*_{2n+1},4*F*_{2n+3},*F*_{2n+7}), *B*=(*F*_{2n+1},4*F*_{2n+5},*F*_{2n+7}), *C*=(*F*_{2n+1},5*F*_{2n+1},*F*_{2n+3}), *D*=(*F*_{2n+3},4*F*_{2n+1},*F*_{2n+3}) and A=(*L*_{2n+1},4*L*_{2n+3},*L*_{2n+7}), B=(*L*_{2n+1},4*L*_{2n+5},*L*_{2n+7}), C=(*L*_{2n+1},5*L*_{2n+1},*L*_{2n+3}), D=(*L*_{2n+3},4*L*_{2n+1},*L*_{2n+3}. The sequences *A*,*B*,*C* and *D* are built from the Fibonacci numbers *F*_{n} while the sequences A, B, C and D from the Lucas numbers *L*_{n}. 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.

We study the eight infinite sequences of triples of natural numbers

KEYWORDS

Fibonacci Numbers, Lucas Numbers, Square, Symmetric Sum, Alternating Sum, Product, Component

Fibonacci Numbers, Lucas Numbers, Square, Symmetric Sum, Alternating Sum, Product, Component

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.

nullZ. Čerin, "Squares from

References

[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. http://www.research.att.com/ 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.

[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. http://www.research.att.com/ 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.