Let be an odd prime. By using the elementary methods we prove that: (1) if , the Diophantine equation has no positive integer solution except when or is of the form , where is an odd positive integer. (2) if , , then the Diophantine equation has no positive integer solution.
Let f ∈ ℚ [X] and deg f ≤ 3. We prove that if deg f = 2, then the diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in ℚ (t). In the case when deg f = 3 and f(X) = X(X²+aX+b) we show that for all but finitely many a,b ∈ ℤ satisfying ab ≠ 0 and additionally, if p|a, then p²∤b, the equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in rationals.
Let denote the term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation in the positive integers and , where and are given positive integers. A complete solution is given if the exponents are included in the set . Based on the specific cases we could solve, and a computer search with we conjecture that beside the trivial solutions only , , and satisfy the title equation.