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 $$ax + by =c$$ is a simple indeterminate equation, as is $$x^2=1$$. 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:
 * $$a_nx^n+a_{n-1}x^{n-1}+\dots +a_2x^2+a_1x+a_0 = 0,$$

which has multiple solutions for the variable $$x$$ in the complex plane—unless it can be rewritten in the form $$a_n(x-b)^n = 0$$.

Non-degenerate conic equation:


 * $$Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0,$$

where at least one of the given parameters $$A$$, $$B$$, and $$C$$ is non-zero, and $$x$$ and $$y$$ are real variables.

Pell's equation:
 * $$\ x^2 - Py^2 = 1,$$

where $$P$$ is a given integer that is not a square number, and in which the variables $$x$$ and $$y$$ are required to be integers.

The equation of Pythagorean triples:
 * $$x^2+y^2=z^2,$$

in which the variables $$x$$, $$y$$, and $$z$$ are required to be positive integers.

The equation of the Fermat–Catalan conjecture:
 * $$a^m+b^n=c^k,$$

in which the variables $$a$$, $$b$$, $$c$$ are required to be coprime positive integers, and the variables $$m$$, $$n$$, and $$k$$ are required to be positive integers satisfying the following equation:
 * $$\frac{1}{m} + \frac{1}{n} + \frac{1}{k} < 1. $$