The Integral Points On A Class Of Elliptic Curve

Abstract

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 y 2 = x 3 + a x + b, a, b ∈ Z. In 1987, D. Zagier proposed the integer points problem on y 2 = x 3 - 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 y 2 = x 3 - 27x + 62. In 2010, Wu H M used the elementary methods to find all the integral points of elliptic curves y 2 = x 3 - 27x - 62. In 2015, Li Y Z and Cui B J used the elementary methods to solve the problem of the integer points on y 2 = x 3 - 21x - 90. In 2016, Guo J used the elementary methods to solve the problem of the integer points on y 2 = x 3 + 27x + 62. In 2017, Guo J used the elementary method proves that y 2 = x 3 - 21x + 90 has no integer points. Up to now, there is no relevant conclusions on the integral points of elliptic curves y 2 = x 3 + 19x - 46, which is the subject of this paper. By using congruence and Legendre Symbol, it can be proved that elliptic curve y 2 = x 3 + 19x - 46 has only one integer point: (x, y) = (2,0).