Indeterminate equation

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In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution.[1] For example, the equation is a simple indeterminate equation, as is . 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:

which has multiple solutions for the variable in the complex plane—unless it can be rewritten in the form .

Non-degenerate conic equation:

where at least one of the given parameters , , and is non-zero, and and are real variables.

Pell's equation:

where is a given integer that is not a square number, and in which the variables and are required to be integers.

The equation of Pythagorean triples:

in which the variables , , and are required to be positive integers.

The equation of the Fermat–Catalan conjecture:

in which the variables , , are required to be coprime positive integers, and the variables , , and are required to be positive integers satisfying the following equation:

See also

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References

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  1. ^ "Indeterminate Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2019-12-02.
  2. ^ "Indeterminate Equation – Lexique de mathématique". 12 October 2018. Retrieved 2019-12-02.