Elliptic Curve Integral Points on y2 = x3 + 3x ? 14

3rd International Conference on Energy Equipment Science and Engineering

Zhao, Jianhong^1

Department of Teachers and Education, Lijiang Teachers College, Lijiang, Yunnan

674199, China^1

关键词: Algebraic number theory;    Arithmetic property;    Elementary methods;    Elliptic curve;    Integer point;    Legendre;    P-Adic analysis;    Positive integers;   

Others  :  https://iopscience.iop.org/article/10.1088/1755-1315/128/1/012108/pdf
DOI  :  10.1088/1755-1315/128/1/012108

来源: IOP

【 摘 要 】

The positive integer points and integral points of elliptic curves are very important in the theory of number and arithmetic algebra, it has a wide range of applications in cryptography and other fields. There are some results of positive integer points of elliptic curve y2= x3+ ax + b, a, b ∈ Z In 1987, D. Zagier submit the question of the integer points on y2= x3- 27x + 62, it count a great deal to the study of the arithmetic properties of elliptic curves. In 2009, Zhu H L and Chen J H solved the problem of the integer points on y2= x3- 27x + 62 by using algebraic number theory and P-adic analysis method. In 2010, By using the elementary method, Wu H M obtain all the integral points of elliptic curves y2= x3- 27x - 62. In 2015, Li Y Z and Cui B J solved the problem of the integer points on y2= x3- 21x - 90 By using the elementary method. In 2016, Guo J solved the problem of the integer points on y2= x3+ 27x + 62 by using the elementary method. In 2017, Guo J proved that y2= x3- 21x + 90 has no integer points by using the elementary method. Up to now, there is no relevant conclusions on the integral points of elliptic curves y2= x3+ 3x - 14, which is the subject of this paper. By using congruence and Legendre Symbol, it can be proved that elliptic curve y2= x3+ 3x - 14 has only one integer point: (x, y) = (2, 0).