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A note on Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

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.

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 , ± 154914585540 ) , using elementary number theory methods and some known results on quadratic and quartic Diophantine equations.

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