Integral solutions of <mml:math alttext="$x^3  + \lambda y + 1 - xyz = 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>&#955;</mml:mi><mml:mi>y</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>&#8722;</mml:mo><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> are observed for all integral <mml:math alttext="$\lambda $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#955;</mml:mi></mml:math>. For <mml:math alttext="$\lambda  = 2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#955;</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> the 13 solutions of the equation in positive integers are determined. Solutions of the equation in positive integers were previously determined for the case <mml:math alttext="$\lambda  = 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#955;</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>.

International Journal of Mathematics and Mathematical Sciences

Positive solutions of the diophantine equation

Abstract

Copyright