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Using the elementary method and some properties of the least solution of Pell's equation, we prove that the equation x(y) + y(x) = z(z) has no positive integer solutions (x, y, z) with x and y being odd primes.

abstract and applied analysis

On the Odd Prime Solutions of the Diophantine Equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msup></mml:math>

On the Odd Prime Solutions of the Diophantine Equationxy+yx=zz