On the Diophantine equation (p-1)^x-p^y=z^2, when p is a prime
Keywords:
Diophantine equation, Congruence, Quadratic residueAbstract
In this paper, the non-negative integer solutions of the Diophantine equation
, when
is a prime, are investigated. The results of this research, we showed that if
, then the non-negative integer solutions of the equation are
, when
is a non-negative integer. If
, then the equation has the unique non-negative integer solution, which is
. If
, then all non-negative integer solutions
of the equation are
and
. Moreover, if
and
, then the non-negative integer solutions of the equation are
and
, when
is an integer.
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