We find all positive integer solutions <mml:math alttext="$(x,y,a,b,n)$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> of
<mml:math alttext="$x^2+2^a\cdot 3^b=y^n$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>a</mml:mi></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mn>3</mml:mn><mml:mi>b</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math> with <mml:math alttext="$n\geq 3$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:math> and coprime <mml:math alttext="$x$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi></mml:math> and <mml:math alttext="$y$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>y</mml:mi></mml:math>.

On the equation <mml:math alttext="$x^2+2^a\cdot 3^b=y^n$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>a</mml:mi></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mn>3</mml:mn><mml:mi>b</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math>

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International Journal of Mathematics and Mathematical Sciences

On the equation x2+2a⋅3b=yn

Abstract

Copyright