Indeterminate equation explained

In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution.^[1] 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.^[2] Some of the prominent examples of indeterminate equations include:

Univariate polynomial equation:

a ^{n + a} _{n -1} x ^{n-1} +... + a ^{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} _{n(x - b)} = 0 .

Non-degenerate conic equation:

Ax ^{2} + Bxy + Cy ^{2} + Dx + Ey + F = 0 , 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:(1)/(m) + (1)/(n) + (1)/(k) < 1 .

See also

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

References

1. Web site: Indeterminate Definition (Illustrated Mathematics Dictionary). www.mathsisfun.com. 2019-12-02.
2. Web site: Indeterminate Equation – Lexique de mathématique. 12 October 2018. en-US. 2019-12-02.