Feasibility of Primality in Bounded Arithmetic

We prove the correctness of the AKS algorithm \cite{AKS} within the bounded arithmetic theory $T^{count}_2$ or, equivalently, the first-order consequences of the theory $VTC^0$ expanded by the smash function, which we denote by $VTC^0_2$. Our approach initially demonstrates the correctness within the theory $S^1_2 + iWPHP$ augmented by two algebraic axioms and then show that they are provable in $VTC^0_2$. The two axioms are: a generalized version of Fermat’s Little Theorem and an axiom adding a new function symbol which injectively maps roots of polynomials over a definable finite field to numbers bounded by the degree of the given polynomial. To obtain our main result, we also give new formalizations of parts of number theory and algebra: $\bullet$ In $PV_1$: We formalize Legendre’s Formula on the prime factorization of $n!$, key properties of the Combinatorial Number System and the existence of cyclotomic polynomials over the finite fields $\mathbb{Z}/p$. $\bullet$ In $S^1_2$: We prove the inequality $lcm(1,\dots, 2n) \geq 2^n$. $\bullet$ In $VTC^0$: We verify the correctness of the Kung–Sieveking algorithm for polynomial division.

💡 Research Summary

The paper establishes that the deterministic polynomial‑time AKS primality‑testing algorithm can be proved correct within the weak bounded‑arithmetic theory (T^{count}_2), equivalently within the first‑order consequences of (VTC^0) augmented by the smash function (denoted (VTC^0_2)). The authors proceed in two main stages. First, they show that the correctness of AKS is provable in the theory (S^1_2) extended by the injective weak pigeonhole principle (iWPHP) together with two new algebraic axiom schemata: (1) a Generalized Fermat’s Little Theorem (GFL T) and (2) a Root Upper Bound (RUB) axiom. GFL T guarantees that for a logarithmic‑size exponent (r) the congruence ((X+a)^p \equiv X^p + a \pmod{X^r-1}) holds, which is essential for the polynomial‑power step of AKS. RUB introduces a new function symbol that injectively maps the roots of any sparse polynomial (f) over a definable finite field to the set ({1,\dots,\deg f}); this sidesteps the need for an explicit enumeration of roots while preserving the combinatorial structure required by the algorithm.

The second stage proves that the two added axioms are themselves derivable in (VTC^0_2). To achieve this, the authors formalize a collection of auxiliary number‑theoretic and algebraic facts in three sub‑theories:

• In (PV_1) they formalize Legendre’s formula for the prime factorisation of (n!), develop the combinatorial number system, and prove the existence of cyclotomic polynomials over finite fields (\mathbb{Z}/p). These results provide the arithmetic backbone needed for later lemmas.
• In (S^1_2) they prove the inequality (\operatorname{lcm}(1,\dots,2n) \ge 2^n), a key bound used in the analysis of the AKS algorithm’s runtime.
• In (VTC^0) they verify the correctness of the Kung–Sieveking algorithm for polynomial division, showing that high‑degree polynomial division can be carried out within the resources of (VTC^0).

With these components in place, the paper demonstrates that the full AKS correctness proof can be carried out in (S^1_2 + iWPHP + GFL T + RUB), and that this theory is a conservative extension of (VTC^0_2). Consequently, the Σ₈‑consequences of (T^{count}_2) (or equivalently the Σ₁ᴮ‑consequences of (VTC^0_2)) prove the statement \