  We prove general sufficient and necessary conditions for the partition
regularity of Diophantine equations, which extend the classic Rado's Theorem by
covering large classes of nonlinear equations. Sufficient conditions are
obtained by exploiting algebraic properties in the space of ultrafilters betaN,
grounding on combinatorial properties of positive density sets and IP sets.
Necessary conditions are proved by a new technique in nonstandard analysis,
based on the use of the relation of u-equivalence for the hypernatural numbers
*N.
