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 [math]\displaystyle{ ax + by =c }[/math] is a simple indeterminate equation, as is [math]\displaystyle{ x^2=1 }[/math]. 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:
- [math]\displaystyle{ a_nx^n+a_{n-1}x^{n-1}+\dots +a_2x^2+a_1x+a_0 = 0, }[/math]
which has multiple solutions for the variable [math]\displaystyle{ x }[/math] in the complex plane—unless it can be rewritten in the form [math]\displaystyle{ a_n(x-b)^n = 0 }[/math].
Non-degenerate conic equation:
- [math]\displaystyle{ Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0, }[/math]
where at least one of the given parameters [math]\displaystyle{ A }[/math], [math]\displaystyle{ B }[/math], and [math]\displaystyle{ C }[/math] is non-zero, and [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are real variables.
- [math]\displaystyle{ \ x^2 - Py^2 = 1, }[/math]
where [math]\displaystyle{ P }[/math] is a given integer that is not a square number, and in which the variables [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are required to be integers.
The equation of Pythagorean triples:
- [math]\displaystyle{ x^2+y^2=z^2, }[/math]
in which the variables [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], and [math]\displaystyle{ z }[/math] are required to be positive integers.
The equation of the Fermat–Catalan conjecture:
- [math]\displaystyle{ a^m+b^n=c^k, }[/math]
in which the variables [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], [math]\displaystyle{ c }[/math] are required to be coprime positive integers, and the variables [math]\displaystyle{ m }[/math], [math]\displaystyle{ n }[/math], and [math]\displaystyle{ k }[/math] are required to be positive integers satisfying the following equation:
- [math]\displaystyle{ \frac{1}{m} + \frac{1}{n} + \frac{1}{k} \lt 1. }[/math]
See also
References
 |  |
