It has been shown in [2] that the title equation has infinitely many solutions when p = 2 and also when p = 3. In this article, it is established and demonstrated for each prime p > 3, that the equation has a solution for each and every integer x ≥ 1. We also discuss separately two distinct particular cases of the equation. One is related to the Sophie Germain conjecture, and the other to the Goldbach conjecture.
In [7], it is shown that the Diophantine equation 4 x + 7 y = z 2 has no solutions in non-negative integers. In this paper, investigating all odd powers of 2 with all even values of y, we establish that the title equation has only one solution when x = 2 and y = 2, whereas for all other values x no solutions exist.
Abstract. In this short article, it is established for the title equation: (i) No solutions exist when p = 2. (ii) Exactly two solutions exist when p = 3. In both solutions q is prime. (iii) Exactly two solutions exist for each and every prime p > 3 in which q is composite. Some numerical solutions are also exhibited.
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