Welcome to Open Science
Contact Us
Home Books Journals Submission Open Science Join Us News
On The Integer Solutions of the Diophantine Equations x2+ay2 = zn
Current Issue
Volume 4, 2016
Issue 2 (April)
Pages: 9-16   |   Vol. 4, No. 2, April 2016   |   Follow on         
Paper in PDF Downloads: 66   Since Jul. 19, 2016 Views: 1446   Since Jul. 19, 2016
Ahmet Cihangir, Elementary Mathematics Education Programme, Ahmet Kelesoglu Education Faculty, Necmettin Erbakan University, Konya, Turkey.
Hasan Şenay, Elementary Mathematics Education Programme, Faculty of Education, Mevlana University, Konya, Turkey.
In this study; where the variables x, y and z are integers, a and n positive integers, existence of x, y, z integer solutions’ for each positive n integers of x2+ay2 = zn Diophantine equation was shown by induction. Also change of z was analyzed according to the values which a will take and each situation was exemplified.
Exponential Diophantine Equations, Pythagorean Triangles, Higher Degree Equations
Mahler, K.; Zur Approximation Algebra is Cher Zahler I: Uber Den Gr¨ossten Primteiler Bin¨arer Formen. Math. Ann.1933; 107: 691–730.
Gel’fond, A. O.; Sur La Divisibilit´E De La Diff´erence Des Puissances De Deux Nombres Entiers Par Une Puissance D’un İd´Eal Premier. Mat. Sb. 1940; 7: 7–25.
Terai, N.; The Diophantine Equation ax+by=cz, Proc.Japan Acad. Ser.A Math. Sci. 1994; 70: 22–26.
Cao, Z. F.; A Note On The Diophantine Equation ax+by=cz. Acta Arith.1999; 91: 85–93.
Le, M. H.; On The Diophantine Equation ax+by=cz. J. Changchun Teachers College Ser. Nat. Sci. 1985; 2: 50–62 (in Chinese)
Terai, N.; The Diophantine Equation ax+by=cz. III, Proc. Japan Acad. Ser. A Math. Sci.1996; 72: 20–22.
Terai, N.; Applications Of A Lower Bound For Linear Forms In Two Logarithms To Exponential Diophantine Equations. Acta Arith.1999; 90: 17–35.
Dong, X. L., Cao, Z. F.; The Terai–Je´Smanowicz Conjecture Concerning The Equation ax+by=cz. Chinese Math. Ann. Ser. A 2000; 21(A): 709–714 (in Chinese)
Le, M. H.; A Conjecture Concerning the Pure Exponential Diophantine Equation . Acta Math. Sinica, English Series 2004; 21: 943–948.
Terai N.; The Diophantine Equation ax+by=cz II, Proc. Japan Acad. Math. Sci.1995; 71: 109–110.
Sierpinski W.; Elementary Theory of Numbers, A. Schinzel (Ed.) (second English edition), PWN – Polish Scientific Publishers, New York, 1988.
Cihangir A. ve Şenay H.; ap+bq=cr Diophantine Denkleminin Tamsayı Çözümleri Üzerine, S. Ü. Eğitim Fakültesi Dergisi (Fen Bil.) 1998; 7A: 77– 85.
Cihangir A. ve Şenay H.; ap+bq=cr Diophantine Denkleminin Tamsayı Çözümleri Üzerine II, S. Ü. Eğitim Fakültesi Fen Dergisi 2000; 8(1): 137–143.
Cihangir, A. and Şenay, H.; On the Integer Solutions of The Diophantine Equations x2+y2=z2 and xn+y2=z2, Jour. of. Math.& Comp.Sci (Math.Ser.) 2002;15(2):79–84.
Cihangir A; (1998), Pythagorean Üçlüleri Grubu ile z2-ay2=x2 Denkleminin Çözüm Üçlüleri Kümesinin Cebirsel Özellikleri ve xp+ay2=zq Diophantine Denkleminin Tamsayı Çözümleri Üzerine, Ph.D. Thesis, S.Ü. Fen Bilimleri Enstitüsü–Konya, Turkey.
Open Science Scholarly Journals
Open Science is a peer-reviewed platform, the journals of which cover a wide range of academic disciplines and serve the world's research and scholarly communities. Upon acceptance, Open Science Journals will be immediately and permanently free for everyone to read and download.
Office Address:
228 Park Ave., S#45956, New York, NY 10003
Phone: +(001)(347)535 0661
Copyright © 2013-, Open Science Publishers - All Rights Reserved