Yagel Numbers: Primorial Generalization of Mersenne Numbers
Abstract
Yagel numbers represent a novel generalization of Mersenne numbers, constructed using primorials, the product of consecutive prime numbers, to amplify prime density. Defined as:
$$ Y_k(n) = \left(\prod_{p=1}^{k-1} primes \right) \cdot p_k^n - 1, \quad n,k \in \mathbb{N}^* $$
Where \( p_k \) is the \( k \)-th prime raised to the power \( n \).
These numbers systematically exclude divisibility by small primes, creating a refined search space for primes. Inspired by the sieve-like behavior of Mersenne numbers, Yagel numbers extend this idea to higher orders, revealing structural biases that favor primality.
This paper explores Yagel numbers’ growth patterns, prime densities, and computational feasibility. Through numerical experiments and probabilistic primality testing (Miller-Rabin), Yagel numbers exhibit an unexpectedly high density of primes within specific ranges. While their exponential growth imposes practical limitations for large-scale searches, their observed deviations from expected prime densities raise theoretical questions about prime structures and sieving mechanisms.
Rather than competing with established methods like the Lucas-Lehmer test for Mersenne primes, Yagel numbers offer a thought experiment – combining intuition and computation to examine prime generation through structured filtering. This exploration aims to inspire further research into prime-rich constructs and sieve-based strategies for prime discovery.
Introduction
This paper introduces Yagel numbers, a natural generalization of Mersenne numbers designed to amplify prime density using primorial factors. A Yagel number of order \( k \) and exponent \( n \), denoted as \( Y \), where \( k \) and \( n \) are positive natural numbers, is defined as:
$$ Y_k\left(n\right)=p_{k-1}\#\cdot p_k^n-1=\left(\prod_{p=1}^{k-1}primes\right)\cdot p_k^n-1=2\cdot3\cdot5\cdot7\cdot11\cdot\ \cdots\ \ \cdot p_{k-1}\cdot p_k^n-1 $$
where the product includes all primes up to the \( (k-1) \)-th prime and \( p_k^n \) is the \( k \)-th prime raised to the power \( n \). Here, the order parameter \( k \) refers to the number of prime factors used in the scalar factor, or primorial, while \( n \) represents the exponent of the \( k \)-th prime. This generalization builds directly on the structure of Mersenne numbers, which are a special case and the first-order Yagel numbers, where:
$$ M_n=Y_1\left(n\right)=2^n-1 $$
Full Paper
The full paper, including detailed explanations, proofs, and computational results, can be viewed or downloaded below: