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Paper IPM / M / 8009  


Abstract:  
Let p be an odd prime such that p−3 is not a perfect square.
In this paper we prove that the equation x^{2}+3=py^{p−1} has no
solutions in rational numbers x,y. The proof depends on the
unique factorization in the ring of algebraic integers of
\mathbbQ(√−3) and on certain congruence arguments.
Furthermore, the equations x^{2}+3=py^{[(p−1)/2]} and
x^{2}+3=py^{6} in rationals x,y are also considered.
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