Diophantine Equation - Examples of Diophantine Equations

Examples of Diophantine Equations

In the following Diophantine equations, x, y, and z are the unknowns, the other letters being given are constants.
This is a linear Diophantine equation (see the section "Linear Diophantine equations" below).
For n = 2 there are infinitely many solutions (x,y,z): the Pythagorean triples. For larger integer values of n, Fermat's Last Theorem states there are no positive integer solutions (x, y, z).
(Pell's equation) which is named after the English mathematician John Pell. It was studied by Brahmagupta in the 7th century, as well as by Fermat in the 17th century.
The Erdős–Straus conjecture states that, for every positive integer n ≥ 2, there exists a solution in x, y, and z, all as positive integers. Although not usually stated in polynomial form, this example is equivalent to the polynomial equation 4xyz = yzn + xzn + xyn = n(yz + xz + xy).

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