2025/08/15 by Alpay, Faruk, Alpay, Taylan
#03C64 #11A41 #11N05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2508.11466
A single closed expression fAlpay,U(x) is presented which, for every integer x≥ 0, returns the (x+1)-th prime px+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)=∑j≤ iI(j) equals π(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≥ 2, the integer P^⋆=fAlpay,U(L) is prime and P^⋆>L. Two explicit schedules U(x) are given: a square schedule Usq(x)=(x+1)2 and a near-linear schedule UAlpay-lin(x)=Θ(xlog x) justified by explicit bounds on pn. 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)=Ω(xlog x) while UAlpay-lin achieves O(xlog x). We also provide explicit operation counts (''form complexity'') of the folded expression.