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Harvey Friedman asked in 1986 whether the function f(x,y) = x^2 + y^3 on thereal plane R^2 satisfies any identities; examples of identities arecommutativity and associativity. To solve this problem of Friedman, we musteither find a nontrivial identity involving expressions formed by recursivelyapplying f to a set of variables {x_1,x_2, ..., x_n} that holds in the realnumbers or to prove that no such identities hold. In this paper, we will solvecertain special cases of Friedman's problem and explore the connection betweenthis problem and certain Diophantine equations.

Identities of the Function f(x,y) = x^2 + y^3