Elliptic Curve Integral Points on y2 = x3 + 3x − 14
Jianhong Zhao · IOP Conference Series Earth and Environmental Science · 2018
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 y 2 = x 3 + ax + b , a , b ∈ Z In 1987, D. Zagier submit the question of the integer points on y 2 = x 3 − 27 x + 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 y 2 = x 3 − 27 x + 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 y 2 = x 3 − 27 x − 62. In 2015, Li Y Z and Cui B J solved the problem of the integer points on y 2 = x 3 − 21 x − 90 By using the elementary method. In 2016, Guo J solved the problem of the integer points on y 2 = x 3 + 27 x + 62 by using the elementary method. In 2017, Guo J proved that y 2 = x 3 − 21 x + 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 y 2 = x 3 + 3 x − 14, which is the subject of this paper. By using congruence and Legendre Symbol, it can be proved that elliptic curve y 2 = x 3 + 3 x − 14 has only one integer point: ( x, y ) = (2, 0).