Indeterminate equation

In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution.[1] For example, the equation ${\displaystyle ax+by=c}$ is a simple indeterminate equation, as is ${\displaystyle x^{2}=1}$. Indeterminate equations cannot be solved uniquely. In fact, in some cases it might even have infinitely many solutions.[2] Some of the prominent examples of indeterminate equations include:

Univariate polynomial equation:

${\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{2}x^{2}+a_{1}x+a_{0}=0,}$

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

Non-degenerate conic equation:

${\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,}$

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

Pell's equation:

${\displaystyle \ x^{2}-Py^{2}=1,}$

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

The equation of Pythagorean triples:

${\displaystyle x^{2}+y^{2}=z^{2},}$

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

The equation of the Fermat–Catalan conjecture:

${\displaystyle a^{m}+b^{n}=c^{k},}$

in which the variables ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ are required to be coprime positive integers, and the variables ${\displaystyle m}$, ${\displaystyle n}$, and ${\displaystyle k}$ are required to be positive integers satisfying the following equation:

${\displaystyle {\frac {1}{m}}+{\frac {1}{n}}+{\frac {1}{k}}<1.}$