ON THE SOME PARTICULAR SETS

Year 2016, Volume: 2 Issue: 2, 99 - 108, 25.12.2016

Özen Özer

Abstract

For 𝑡 an integer, a 𝑃𝑡 set is defined as a set of 𝑚 positive integers with the property that the product of its any two distinct element increased by 𝑡 is a perfect square integer.

In this study, the certain special 𝑃−5, 𝑃+5, 𝑃−7 and 𝑃+7 sets with size three are considered. It is demonstrated that they cannot be extended to 𝑃−5, 𝑃+5, 𝑃−7 and 𝑃+7 with size four. Also, some properties of them are proved.

Keywords

𝑃𝑡 Sets, Congruences, Reciprocity

References

• [1] Anglin W.S., The queen of mathematics-An introduction to number theory, Kluwer Academic Publishers, Dordrecht, 1995.
• [2] Baker A., Davenport H., 1969, The equations 3x2−2=y2 and 8x2−7=z2 , Quarterly Journal of Mathematics, Oxford(2), 20, 129-137, 1969.
• [3] Brown LE., Sets in Which xy+k is Always a Square, Math. Comp, 45, 613-620, 1985.
• [4] Dickson LE., History of Theory of Numbers and Diophantine Analysis, Vol 2, Dove Publications, New York, 2005.
• [5] Filipin L A., Fujita Y., M. Mignotte, The non-extendibility of some parametric families of D(-1)-triples, Q. J. Math. 63, 605-621, 2012.
• [6] Gopalan M. A., Vidhyalakshmi S., Mallika S., Some special non-extendable Diophantine triples, Sch. J. Eng. Tech. 2, 159-160, 2014.
• [7] Grinstead C.M., On a Method of Solving a Class of Diophantine Equations . Math. Comp.,32, 936-940, 1978.
• [8] Kanagasabapathy P., Ponnudurai T., The Simultaneous Diophantine Equations y2−3x2=−2 and z2−8x2=−7 , Quarterly Journal of Mathernatics, Oxford Ser (3), 26, 275-278, 1975.
• [9] Katayama S., Several methods for solving simultaneous Fermat-Pell equations, J. Math. Tokushima Univ., 33 , 1–14, 1999.
• [10] Kaygısız K., Şenay H., Contructions of Some New Nonextandable Pk Sets, International Mathematical Forum, 2, no. 58, 2869 – 2874, 2007.
• [11] Masser D.W., Rickert J.H., Simultaneous Pell Equations ,Number Theory , 61, 52-66, 1996.
• [12] Mohanty P., Ramasamy A.M.S., The Simultaneous Diophantine Equations 5y2−20= x2,2y2+1= z2,.J.Number Theory.18,365-359, 1984.
• [13] Mordell LJ., Diophantine Equations, Academic Press, New York, 1970.
• [14] Mollin R.A., Fundamental Number theory wiyh Applications, CRC Press, 2008.
• [15] Tzanakis N., Effective solution of two simultaneous Pell equations by the elliptic logarithm method, Acta Arithm., 103, 119–135, 2002.