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An indeterminate equation, in mathematics, is an equation for which there is more than one solution; for example, 2x = y is a simple indeterminate equation, as are \( ax + by = c \) and \( x^2 = 1 \). Indeterminate equations cannot be solved uniquely. Prominent examples include the following:

Univariate polynomial equation:

\( a_nx^n+a_{n-1}x^{n-1}+\dots +a_2x^2+a_1x+a_0 = 0, \)

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 the sum of whose reciprocals is less than 1.

See also

Indeterminate system
Indeterminate (variable)
Linear algebra

  • Mathematics Encyclopedia

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