The main purpose of this paper is to prove that the elliptic curve $E:y^2=x^3+27x-62$ has only the integral points $(x,y)=(2,0)$ and $(28844402,\pm 154914585540)$, using elementary number theory methods and some known results on quadratic and quartic Diophantine equations.

A Diophantine $m$-tuple is a set of $m$ positive integers such that the product of any two of them is one less than a perfect square. In this paper we study some properties of elliptic curves of the form $y^2=(ax+1)(bx+1)(cx+1)$, where $\{a,b,c\}$, is a Diophantine triple. In particular, we consider the elliptic curve $E_k$ defined by the equation $y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1),$ where $k\ge 2$ and $F_n$, denotes the $n$-th Fibonacci number. We prove that if the rank of $E_k\left(𝐐\right)$ is equal to one, or $k\le 50$, then all integer points on $E_k$ are given by $$\begin{array}{c}(x,y)\in \{(0\pm 1),(4F_{2k+1}{F}_{2k+2}{F}_{2k+3}\pm \left(2F_{2k+1}{F}_{2k+2}-1\right)\hfill \\ \hfill \times \left(2F_{2k+2}^2+1\right)\left(2F_{2k+2}{F}_{2k+3}+1\right)\}.\end{array}$$

This paper investigates the system of equations $$x^2+a{y}^m=z_1^2,x^2-a{y}^m=z_2^2$$ in positive integers $x$, $y$, $z_1$, $z_2$, where $a$ and $m$ are positive integers with $m\ge 3$. In case of $m=2$ we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient $a$ with the condition $gcd(x,z_1)=gcd(x,z_2)=gcd(z_1,z_2)=1$. Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero $a$.

Let k ∈ ℤ be such that ${|}_k\left(\mathbb{Q}\right)|$ is finite, where ${}_k:y²=1-2kx+k²x²-4x³$. We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.

Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height $h{̂}_{E_{\omega }}$ of the specialized elliptic curve $E_{\omega }$ with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient $\left(h{̂}_{E_{\omega }}\left(P_{\omega }\right)\right)/h\left(\omega \right)$ over all nontorsion P ∈ E(K).

Let k be a real quadratic field and let ${}_k$ and ${}_k^{\times }$ be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v ($X{\in }_k$, $u,v{\in }_k^{\times }$) is developed.

We completely solve the Diophantine equations $x²+2^a{q}^b=yⁿ$ (for q = 17, 29, 41). We also determine all $C=p{₁}^{a₁}\cdots p_k^{a_k}$ and $C=2^{a₀}p{₁}^{a₁}\cdots p_k^{a_k}$, where $p₁,...,p_k$ are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).