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:
:$a_nx^n+a_x^+\dots +a_2x^2+a_1x+a_0 = 0,$

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:
: $\frac + \frac + \frac < 1.$

See also

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

References

Algebra

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