Indeterminate equation

In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation formula_1 is a simple indeterminate equation, as are formula_2 and formula_3. Indeterminate equations cannot be solved uniquely. In fact, in some cases it might even have infinitely many solutions. Some of the prominent examples of indeterminate equations include:
Univariate polynomial equation:
which has multiple solutions for the variable formula_5 in the complex plane—unless it can be rewritten in the form formula_6.
Non-degenerate conic equation:
where at least one of the given parameters formula_8, "formula_9", and "formula_10" is non-zero, and "formula_5" and "formula_12" are real variables.
Pell's equation:
where "formula_14" is a given integer that is not a square number, and in which the variables "formula_5" and "formula_12" are required to be integers.
The equation of Pythagorean triples:
in which the variables "formula_5", "formula_12", and "formula_20" are required to be positive integers.
The equation of the Fermat–Catalan conjecture:
in which the variables "formula_22", "formula_23", "formula_24" are required to be coprime positive integers, and the variables "formula_25", "formula_26", and "formula_27" are required to be positive integers satisfying the following equation: