If you accept polynomials in more than one variable...

There is a polynomial inequality in 26 variables that describes the set of primes  (Wikipedia - from where I also sourced the rest of this answer). By a result of Matiyasevich there is another polynomial inequality in 10 variables that also describes the primes, but whose order in on the order of $10^{45}$, which dwarfs Wadim's answer. In the other direction there is an order 4 polynomial inequality that will do the job, but in 58 variables.

If you accept polynomials in more than one variable...

There is a polynomial inequality in 26 variables that describes the set of primes:

(source: Wikipedia). By a result of Matiyasevich there is another polynomial inequality in 10 variables that also describes the primes, but whose order is approximately $10^{45}$, which dwarfs Wadim's answer. In the other direction there is an order 4 polynomial inequality that will do the job, but in 58 variables.

If you accept polynomials in more than one variable...

There is a polynomial inequality in 26 variables that describes the set of primes (Wikipedia - from where I also sourced the rest of this answer). By a result of Matiyasevich there is another polynomial inequality in 10 variables that also describes the primes, but whose order in on the order of $10^{45}$, which dwarfs Wadim's answer. In the other direction there is an order 4 polynomial inequality that will do the job, but in 58 variables.

If you accept polynomials in more than one variable...

There is a polynomial inequality in 26 variables that describes the set of primes (Wikipedia - from where I also sourced the rest of this answer). By a result of Matiyasevich there is another polynomial inequality in 10 variables that also describes the primes, but whose order in on the order of $10^{45}$, which dwarfs Wadim's answer. In the other direction there is an order 4 polynomial inequality that will do the job, but in 58 variables.