The Integral Points On A Class Of Elliptic Curve

• Published in: Journal of physics. Conference series Vol. 1634; no. 1; pp. 12102 - 12107
• Main Author: Yang, Lixing
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 01.09.2020
• Subjects: Curves; Integers; Integrals; Number theory; Physics; Yttrium
• ISSN: 1742-6588; 1742-6596
• DOI: 10.1088/1742-6596/1634/1/012102

It is very important to study the integer points of various elliptic curves in elementary number theory. On the one hand, as an important research object in elementary number theory, elliptic curve plays an indispensable role in the development of mathematics. On the other hand, as an important research object in elementary number theory, elliptic curve has practical application in many aspects. So far, there are some conclusions about elliptic curve y2 = x3 + ax + b, a, b ∈ Z. In 1987, D. Zagier proposed the integer points problem on y2 = x3 - 27x + 62, which is important for studying the arithmetic and geometric properties of elliptic curves. In 2009, Zhu H L and Chen J H used the methods of P-adic analysis and algebraic number theory to solve the problem of the integer points on y2 = x3 - 27x + 62. In 2010, Wu H M used the elementary methods to find all the integral points of elliptic curves y2 = x3 - 27x - 62. In 2015, Li Y Z and Cui B J used the elementary methods to solve the problem of the integer points on y2 = x3 - 21x - 90. In 2016, Guo J used the elementary methods to solve the problem of the integer points on y2 = x3 + 27x + 62. In 2017, Guo J used the elementary method proves that y2 = x3 - 21x + 90 has no integer points. Up to now, there is no relevant conclusions on the integral points of elliptic curves y2 = x3 + 19x - 46, which is the subject of this paper. By using congruence and Legendre Symbol, it can be proved that elliptic curve y2 = x3 + 19x - 46 has only one integer point: (x, y) = (2,0).