There is a recursive definition of the set of squares which uses only addition:
$CS(x,y) := IS(x) \wedge IS(y) \wedge x \lt y \wedge \forall z: (x \lt z) \wedge (z \lt y)⇒\neg IS(z)$
$IS(x)⇔ x=0 \vee x=1 \vee \exists y: \exists z: (CS(z,y) \wedge IS(y) \wedge (\forall z: (y < z \wedge z < x) ⇒ \neg IS(z)) \wedge x + z = y + y + 2)$
($IS$ stands for IsSquare, $CS$ stands for ConsecutiveSquares.)
Expanding $CS$ in terms of $IS$ gives an axiom in the language:
$IS(x) ⇔x=0 \vee x=1 \vee
\exists y: \exists z: (IS(z) \wedge IS(y) \wedge z \lt y \wedge \forall w: (z \lt w) \wedge (w \lt y)⇒\neg IS(w)) \wedge IS(y) \wedge
(\forall z: (y < z \wedge z < x) ⇒ \neg IS(z)) \wedge
x + z = y + y + 2
This seems to generalize to a recursive definition, using only addition, of the set of values of any univariate polynomial. Formally I am considering Presburger arithmetic with a unary predicate and associated axioms, yielding an incomplete system in this case.
There are other unary predicates which result in a complete theory. But my question is:
Is there any recursive definition, using only addition, of the set of values of $x^2+y^2$? If so, does it yield an incomplete system? Otherwise, how to prove that such a definition does not exist?
I recently asked "Is there no univariate integer polynomial that takes on the same positive values as the multivariate polynomial $x^2+y^2$? and accepted an answer but I don't understand how to apply it to this situation.