## Quick Reference

An algebraic equation in one or more unknowns, with integer coefficients, for which integer solutions are required. A great variety of Diophantine equations have been studied. Some have infinitely many solutions, some have finitely many and some have no solutions. For example:*x*+9*y*=1 has solutions *x*=2+9*t*, *y*=−3−14*t* (where *t* is any integer).*x*^{2}+1=2*y*^{4} has two solutions *x*=1, *y*=1 and *x*=239, *y*=13.*x*^{3}+*y*^{3}=*z*^{3} has no solutions. See also Hilbert's tenth problem and Pell's equation.

*x*+9*y*=1 has solutions *x*=2+9*t*, *y*=−3−14*t* (where *t* is any integer).

*x*^{2}+1=2*y*^{4} has two solutions *x*=1, *y*=1 and *x*=239, *y*=13.

*x*^{3}+*y*^{3}=*z*^{3} has no solutions.

*Subjects:*
Mathematics.

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