Indeterminate equation

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 x² = 1. Indeterminate equations cannot be solved uniquely. Prominent examples include the following:

Univariate polynomial equation:

a_{n}x^{n}+a_{{n-1}}x^{{n-1}}+\dots +a_{2}x^{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 $a_{n}(x-b)^{n}=0$ .

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.

References

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Indeterminate equation

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References