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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