Alpay Folded Prime Enumerator via gcd and Floors: Exact Enumeration, Record-Lift, and Non-Synonymy/Minimality Certificates

TL;DR

This paper introduces an exact prime enumeration formula using gcd and floor functions, providing new certificates and bounds for prime counting and minimality, with explicit operational complexity analysis.

Contribution

It presents a novel closed-form prime enumerator using gcd and floors, along with certificates and bounds for minimality and enumeration efficiency.

Findings

Exact prime enumeration formula using gcd and floors.

Certificates for non-synonymy and minimality bounds.

Explicit complexity analysis of the enumeration expression.

Abstract

A single closed expression $f_{\text{Alpay},U}(x)$ is presented which, for every integer $x\ge 0$, returns the $(x+1)$-th prime $p_{x+1}$. The construction uses only integer arithmetic, greatest common divisors, and floor functions. A prime indicator $I(j)$ is encoded through a short $\gcd$-sum; a cumulative counter $S(i)=\sum_{j\le i}I(j)$ equals $\pi(i)$; and a folded step $A(i,x)$ counts precisely up to the next prime index without piecewise branching. A corollary shows that for any fixed integer $L\ge 2$, the integer $P^\star=f_{\text{Alpay},U}(L)$ is prime and $P^\star>L$. Two explicit schedules $U(x)$ are given: a square schedule $U_{\text{sq}}(x)=(x+1)^2$ and a near-linear schedule $U_{\text{Alpay-lin}}(x)=\Theta(x\log x)$ justified by explicit bounds on $p_n$. We include non-synonymy certificates relative to Willans-type enumerators (schedule and operator-signature separation) and prove an asymptotic minimality bound: any forward-count enumerator requires $U(x)=\Omega(x\log x)$ while $U_{\text{Alpay-lin}}$ achieves $O(x\log x)$. We also provide explicit operation counts (''form complexity'') of the folded expression.