A Note on the Particular Set with Size Three

Özen Özer

Abstract


For a fixed integer k, a Pk  – set is defined as a set of n positive integers { x1, x2, x3,..., xn} with the property that xi, xj + k is a perfect square, whenever i ≠ j . In this paper, we prove P11 = {1,14,25}, P−11 = {1,15,36}, and P−11{4,9,23} sets cannot be extendable. It is also proved that P11 sets don’t contain any multiple of 3 and P−11 sets don’t include any multiple of 7. Moreover, it is demonstrated that all of the elements of the sets P−11 of size three cannot be odd positive integer. Mathematics Subject Classifications: 11A07, 11D45, 11A15.


Keywords:
Congruences, Pk sets, Pell’s equations, Legendre symbol.

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References


W. S. Anglin, The queen of mathematics: An introduction to number theory, Dordrecht: Kluwer Academic Publishers, 1995. https://doi.org/10.1007/978-94-011-0285-8

A. Baker and H. Davenport, “The equations 3x2 – 2 = y2 and 8x2 −7 = z2,” The Quarterly Journal of Mathematics, vol. 20, issue 1, pp. 129–137, 1969. https://doi.org/10.1093/qmath/20.1.129

E. Brown, “Sets in Which xy + k is Always a Square,” Math. Comp., vol. 45, pp. 613–620, 1958.

A. Filipin, Y. Fujita, and M. Mignotte, “The non-extendibility of some parametric families of D(-1)-triples,” Q. J. Math., vol. 63, issue 3, pp. 605–621, 2012. https://doi.org/10.1093/qmath/har013

M. A. Gopalan, S. Vidhyalakshmi and S. Mallika, “Some special non-extendable Diophantine triples,” Sch. J. Eng. Tech., vol. 2, pp. 159–160, 2014.

C. M. Grinstead, “On a Method of Solving a Class of Diophantine Equations,” Math. Comp., vol. 32, pp. 936–940, 1978. https://doi.org/10.1090/S0025-5718-1978-0491480-0

P. Kanagasabapathy and T. Ponnudurai, “The Simultaneous Diophantine Equations y2 – 3x2 = −2 and z2 – 8x2 = −7”, Quarterly Journal of Mathernatics, vol. 26, issue 1, pp. 275–278, Jan. 1975. https://doi.org/10.1093/qmath/26.1.275

S. Katayama, “Several methods for solving simultaneous Fermat-Pell equations,” J. Math. Tokushima Univ., vol. 33, pp. 1–14, 1999.

K. Kaygısız and H. Şenay, “Contructions of Some New Nonextandable Pk Sets,” International Mathematical Forum, vol. 2, no. 58, pp. 2869–2874, 2007.

D. W. Masser and J. H. Rickert, “Simultaneous Pell Equations,” Journal of Number Theory, vol. 61, issue 1, pp. 52–66, Nov. 1996. https://doi.org/10.1006/jnth.1996.0137

P. Mohanty and A. M. S. Ramasamy, “The Simultaneous Diophantine Equations 5y2 – 20 = x2 , 2y2 + 1 = z2”, Journal of Number Theory, vol. 18, issue 3, pp. 365–359, 1984. https://doi.org/10.1016/0022-314X(84)90068-4

R. A. Mollin, Fundamental Number theory with Applications, 2nd ed. CRC Press, 2008. https://doi.org/10.1201/b15895

N. Tzanakis, “Effective solution of two simultaneous Pell equations by the elliptic logarithm method,” Acta Arithm., vol. 103, pp. 119–135, 2002. https://doi.org/10.4064/aa103-2-2




DOI: 10.7250/bfpcs.2016.009

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