A Generating Formula for Prime Candidates: the Mod-100 Wheel G(n,k) = 100n+k
Year: 2026
Type: preprint
Abstract: We study the closed-form expression $G(n,k)=100n+k$, where $n\ge 0$ and $k$ runs over the 40 residues in $[1,99]$ coprime to 10. We prove that $G$ is a bijection onto the set $\mathcal{W}$ of positive integers coprime to 10, with explicit inverse $x\mapsto(\lfloor x/100\rfloor,\ x \bmod 100)$ computable in constant time. Every prime other than 2 and 5 is therefore generated exactly once, and the non-prime outputs are exactly the integer 1 together with the composites coprime to 10. We verify the classification of all outputs below $10^9$ ($4\cdot10^8$ values, of which 50,847,532 are prime), show that the outputs form a rooted tree of depth $\Omega(x)$, and quantify how the share of primes is replaced by higher levels as $N$ grows: the mean of $\Omega$ over the outputs is $\ln\ln N + B - 5/4 + O(1/\ln N)$ with $B\approx 1.0347$, while the count of primes grows without bound. The primes are distributed among the 40 classes with a relative spread of 0.085% at $10^9$. We introduce the divisibility filter $x \bmod k \ne 0$, $k\in K_{\max}(N)=\{k\le\sqrt{N}:\gcd(k,10)=1\}$, and prove that it yields a generator with precision and recall 1 on $[1,N]$; with the fixed set $k\le 99$ the precision is 1 for $N\le 10{,}200$ and above 99% for $N\le 12{,}826$. Using precision, recall and $F_1$ we compare the variants with wheels of other moduli and with the formulas of Mills and Willans, and we report measured running times, per-output costs and memory footprints.
Author: Mohammed, Ahmed Gomaa
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225231997
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