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:

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

which has multiple solutions for the variable $x$ 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:

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

See also

• Indeterminate form
• Indeterminate system
• Indeterminate (variable)
• Linear algebra