Jianhong Zhao

By using elementary number theory methods, such as congruence and Legendre Symbol, it can be proved that elliptic curve y² = x³ + 19x + 46 has no integer point.

The main purpose of this paper is applying the analysis method, the properties of Lucas polynomials and Gauss sums to study the estimation problems of some kind hybrid character sums. In the end, we obtain several sharp upper bound estimates for them. As some applications, we prove some new and interesting combinatorial identities.

In recent years, scholars have paid much attention to the problem of solving the diophantine equations x ² − D ₁ y ² = m, (D ₁ ∈ Z ⁺, m ∈ Z) and y ² − D ₂ z ² = n, (D ₂ ∈ Z ⁺, n ∈ Z). At present, there are only a few conclusions on x ² − D ₁ y ² = land y ² − D ₂ z ² = 1, and the conclusions mainly concentrated in the number of solutions and the range of it, see Ref [1] and [3]. For odd numbers D ₂, the integer solution of x ² − D ₁ y ² = land y ² − D ₂ z ² = 4, see Ref [4] - [6]. For even numbers D ₂, the integer solution of (2), see Ref [7] - [11]. Up to now, there is no relevant result on the integer solution of x ² − 27y ² = 1 and y ² − 2 ^p z ² = 49, this paper mainly discusses the integer solution of it.

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 − 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 y ² = x ³ − 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 y ² = x ³ − 27x − 62. In 2015, Li Y Z and Cui B J solved the problem of the integer points on y ² = x ³ − 21x − 90 By using the elementary method. In 2016, Guo J solved the problem of the integer points on y ² = x ³ + 27x + 62 by using the elementary method. In 2017, Guo J proved that y ² = x ³ − 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 y ² = x ³ + 3x − 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 + 3x − 14 has only one integer point: (x, y) = (2, 0).

The integer solution of diophantine equations x ² − D ₁ y ² = m, (D ₁ ∈ Z ⁺, m ∈ Z) and y ² − D ₂ z ² = n, (D ₂ ∈ Z ⁺, n ∈ Z) is a matter of great concern. Researchers study for different m, n and D ₁, D ₂, and obtain some correlation results as follows. When m = 1, n = 1, the diophantine equations turns into x ² − D ₁ y ² = 1 and D ₂ z ² = 1. At present, there are only a few conclusions on it, see Ref [1] and [2]. When m = 1, n = 4, the diophantine equations turns into x ² − D ₁ y ² = 1 and D ₂ z ² = 4. For even numbers D ₁, D ₂, the integer solution see Ref [3] - [9]. When m = 1, n = 16, the diophantine equations turns into x ² − D ₁ y ² = 1 and D ₂ z ² = 16. The previous conclusions see Ref [10]. When m = 1, n = 9, the diophantine equations turns into x ² − D ₁ y ² = 1 and D ₂ z ² = 9. Up to now, there is no relevant result on the integer solution of (5), this paper mainly discusses the integer solution of (5) when D ₁ = 7, D ₂ is an even number.