본문 바로가기
KERD
논문KCI 등재

Diophantine triple with Fibonacci numbers and elliptic curve

박진서
대한수학회 (The Korean Mathematical Society)|2021.07|대한수학회논문집 (Communications of the KMS)|제36권 제3호|pp. 401-411|DOI 10.4134/CKMS.c200247

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

Fibonacci numbersDiophantine $m$-tupleelliptic curve
주제분류수학

초록

A Diophantine $m$-tuple is a set $\{a_1, a_2, \dots, a_m\}$ of positive integers such that $a_ia_j+1$ is a perfect square for all $1\leq i < j \leq m$. Let $E_k$ be the elliptic curve induced by Diophantine triple $\{F_{2k}, 5F_{2k+2}, 3F_{2k}+7F_{2k+2}\}$. In this paper, we find the structure of a torsion group of $E_k$, and find all integer points on $E_k$ under assumption that rank$(E_k(\mathbb{Q}))=1$ and some further conditions.

Abstract

A Diophantine $m$-tuple is a set $\{a_1, a_2, \dots, a_m\}$ of positive integers such that $a_ia_j+1$ is a perfect square for all $1\leq i < j \leq m$. Let $E_k$ be the elliptic curve induced by Diophantine triple $\{F_{2k}, 5F_{2k+2}, 3F_{2k}+7F_{2k+2}\}$. In this paper, we find the structure of a torsion group of $E_k$, and find all integer points on $E_k$ under assumption that rank$(E_k(\mathbb{Q}))=1$ and some further conditions.

참고문헌 (19)

  1. A. Baker, "The equations 3x 2 − 2 = y 2 and 8x 2 − 7 = z 2, Quart", J. Math. Oxford Ser, (2) 20, pp.129~137 (1969)[DOI]
  2. J. H. E. Cohn, "Square Fibonacci Numbers", Fibonacci Quart, 2, pp.109~113 (1964)
  3. J. E. Cremona, ", Algorithms for Modular Elliptic Curves" (1997), Cambridge University Press
  4. A. Dujella, "A proof of the Hoggatt-Bergum conjecture", Proc. Amer. Math. Soc, 127(7), pp.1999~2005 (1999)[DOI]
  5. A. Dujella, "A parametric family of elliptic curves", Acta Arith, 94(1), pp.87~101 (2000)[DOI]
  6. A. Dujella, "Diophantine m-tuples and elliptic curves", J. Th´eor. Nombres Bordeaux, 13(1), pp.111~124 (2001)
  7. A. Dujella, "iophantine quadruples and Fibonacci numbers", Bull. Kerala Math. Assoc, 1(2), pp.133~147 (2004)
  8. A. Dujella, "A generalization of a theorem of Baker and Davenport,", Quart. J. Math. Oxford Ser, (2) 49(195), pp.291~306 (1998)[DOI]
  9. A. Dujella, "Integer points on a family of elliptic curves", Publ. Math. Debrecen, 56(3-4), pp.321~335 (2000)
  10. Yasutsugu Fujita, "The Hoggatt-Bergum Conjecture on $D(-1)$-Triples $\{F_{2k+1}, F_{2k+3}, F_{2k+5}\}$ and Integer Points on the Attached Elliptic Curves", Rocky Mountain Journal of Mathematics, 39(6), pp.1907~1932 (2009), Rocky Mountain Mathematics Consortium[DOI]
  11. B. He, "There is no Diophantine quintuple", Trans. Amer. Math. Soc, 371(9), pp.6665~6709 (2019)[DOI]
  12. V. E. Hoggatt, Jr, "A problem of Fermat and the Fibonacci sequence", Fibonacci Quart, 15(4), pp.323~330 (1977)
  13. D. Husemoller, "Elliptic Curves" (1987), Springer-Verlag
  14. A. Kim, "Square Fibonacci numbers and square Lucas numbers", Asian Res. J. Math, 3(3), pp.1~8 (2017)
  15. A. W. Knapp, "Elliptic curves" (1992), Princeton University Press
  16. J. Morgado, "Generalization of a result of Hoggatt and Bergum on Fibonacci numbers", Portugal. Math, 42(4), pp.441~445 (1984)
  17. Ken Ono, "Euler's concordant forms", Acta Arithmetica, 78(2), pp.101~123 (1996), Institute of Mathematics, Polish Academy of Sciences[DOI]
  18. 박진서, "Integer points on the elliptic curves induced by Diophantine triples", 대한수학회논문집, 35(3), pp.745~757 (2020), 대한수학회[DOI]
  19. "SIMATH manual", Saarbr¨ucken, ~ (1997)

관련 논문AI추천

Integer points on the elliptic curves induced by Diophantine triples박진서대한수학회 · 대한수학회논문집2020 · 제35권 제3호Some conditions on the form of third element from Diophantine pairs and its application박진서, 이준복대한수학회 · 대한수학회지2018 · 제55권 제2호The extendibility of Diophantine pairs with property $D(-1)$박진서강원경기수학회 · 한국수학논문집2020 · 제28권 제3호ON ELLIPTIC CURVES WHOSE 3-TORSION SUBGROUP SPLITS AS μ₃ ⊕ Z/3ZMasaya Yasuda대한수학회 · 대한수학회논문집2012 · 제27권 제3호Fibonacci and Lucas Numbers Associated with Brocard-Ramanujan EquationPrapanpong Pongsriiam대한수학회 · 대한수학회논문집2017 · 제32권 제3호GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx 2 AND wx 2 ∓ 1Refik Keskin대한수학회 · 대한수학회보2014 · 제51권 제4호

이 간행물의 다른 논문 (14건)

A relationship between Cayley-Dickson process and the generalized Study determinantPritta Etriana Putri, Laurence Petrus Wijayapp. 413-422Higher order Apostol-type poly-Genocchi polynomials with parameters $a, b$ and $c$Roberto B. Corcino, Cristina B. Corcinopp. 423-445Global attractor for a class of quasilinear degenerate parabolic equations with nonlinearity of arbitrary orderLe Thi Thuy, Tran Thi Quynh Chi, Nguyen Xuan Tupp. 447-463Differences of differential operators between weighted-type spacesJasbir Singh Manhas, Mohammed Said Al Ghafripp. 465-483Some classes of integral equations of convolutions-pair generated by the Kontorovich-Lebedev, Laplace and Fourier transformsTrinh Tuanpp. 485-494
원문 보기