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:

$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,$

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:

${x}^2-P{y}^2=1,$

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