### 14-term arithmetic progressions on quartic elliptic curves.

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We show that the system of equations ${t}_{x}+{t}_{y}={t}_{p},{t}_{y}+{t}_{z}={t}_{q},{t}_{x}+{t}_{z}={t}_{r}$, where ${t}_{x}=x(x+1)/2$ is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system ${t}_{x}+{t}_{y}={t}_{p},{t}_{y}+{t}_{z}={t}_{q},{t}_{x}+{t}_{z}={t}_{r},{t}_{x}+{t}_{y}+{t}_{z}={t}_{s}$ has infinitely many rational two-parameter solutions.

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.