论文信息 - Integral points on the elliptic curve y 2 = x 3 + 27 x − 62 Open image in new window

Integral points on the elliptic curve y 2 = x 3 + 27 x − 62 Open image in new window

We give a new proof that the elliptic curve y 2 = x 3 + 27 x − 62 Open image in new window has only the integral points ( x , y ) = ( 2 , 0 ) Open image in new window and ( x , y ) = ( 28 , 844 , 402 , ± 15 , 491 , 585 , 540 ) Open image in new window using elementary number theory methods and some properties of generalized Fibonacci and Lucas sequences.

Refik Keskin | Olcay Karaatlı

[1] J. Muskat. Generalized Fibonacci and Lucas sequences and rootfinding methods , 1993 .

[2] J. V. Armitage,et al. Introduction to Number Theory , 1966 .

[3] A. Baker. The Diophantine Equation y2 = ax3+bx2+cx+d , 1968 .

[4] Roel J. Stroeker,et al. Computing all integer solutions of a genus 1 equation , 2001, Math. Comput..

[5] Roel J. Stroeker,et al. On the Elliptic Logarithm Method for Elliptic Diophantine Equations: Reflections and an Improvement , 1999, Exp. Math..

[6] Wenpeng Zhang,et al. An elliptic curve having large integral points , 2010 .

[7] P. Ribenboim. An Algorithm to Determine the Points with Integral Coordinates in Certain Elliptic Curves , 1999 .

[8] M. Mignotte,et al. Sur les carrés dans certaines suites de Lucas , 1993 .

[9] D. Kalman,et al. The Fibonacci Numbers—Exposed , 2003 .

[10] Don Zagier,et al. Large Integral Points on Elliptic Curves , 1987 .

[11] S. Rabinowitz. Algorithmic Manipulation of Fibonacci Identities , 1996 .