We can define prime numbers by the following formula in the language of arithmetic: $$P(x):= x>1 \land \forall y< x(\exists z\leq x(y\cdot z =x)\to y=1) $$

This formula is $\Delta_0$ and its properties can be proved in strong enough systems of arithmetic, but in some sub intuitionistic theories of arithmetic properties of this formula can not be proven. By AKS algorithm prime numbers are $\Delta^b_1$ definable in the language of bounded arithmetics (Actually these numbers are polynomial time decidable). My question is about the complexity of defining prime numbers in the language of arithmetic.

Q1. Is there any $E_1$ formula for defining prime numbers?

Q2. Is there any $E^+_1$ formula for defining prime numbers?

$E^+_1$ formulas are $E_1$ formulas which do not have implication or negation.

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