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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.

A note on the diophantine equation a²x⁴ - By² = 1

Jianhua Chen (2001)

Acta Arithmetica

A note on the diophantine equation $(x^{2} - 1) (y^{2} - 1) = {(z^{2} - 1)}^{2}$

Huaming Wu, Maohua Le (1996)

Colloquium Mathematicae

A note on the integer solutions ofhyperelliptic equations

Maohua Le (1995)

Colloquium Mathematicae

A parametric family of elliptic curves

Andrej Dujella (2000)

Acta Arithmetica

A parametric family of quartic Thue equations

Andrej Dujella, Borka Jadrijević (2002)

Acta Arithmetica

A remark on prime divisors of lengths of sides of Heron triangles.

Gaál, István, Járási, István, Luca, Florian (2003)

Experimental Mathematics

A solution to an “unsolved problem in number theory”.

MacLeod, Allan J. (2003)

Southwest Journal of Pure and Applied Mathematics [electronic only]

About a Diophantine equation.

Savin, Diana (2009)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

Addendum on the equation aX⁴ - bY² = 2

S. Akhtari, A. Togbé, P. G. Walsh (2009)

Acta Arithmetica

Algebraic points on cubic hypersurfaces

D. Coray (1976)

Acta Arithmetica

An effective p-adic analogue of a theorem of Thue III. The diophantine equation $y^{2} = x^{2} + k$

J. Coates (1970)

Acta Arithmetica

An effective Shafarevich theorem for elliptic curves

Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)

Acta Arithmetica

An elliptic curve having large integral points

Yanfeng He, Wenpeng Zhang (2010)

Czechoslovak Mathematical Journal

The main purpose of this paper is to prove that the elliptic curve $E : y^{2} = x^{3} + 27 x - 62$ 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.

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