Yagel Numbers: Primorial Generalization of Mersenne Numbers

Understanding Yagel Numbers

Yagel Numbers represent a novel primorial generalization of Mersenne numbers, constructed using primorials—the product of consecutive prime numbers—to amplify prime density.

$Y_{k} (n) = (\prod_{p = 1}^{k - 1} p r i m e s) \cdot p_{k}^{n} - 1, n, k \in N^{*}$

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.

Mersenne primes, which follow the form $M (n) = 2^{n} - 1$ , can be seen as a special case of Yagel Numbers where $k = 1$ . For more information on Mersenne primes, visit Mersenne.org.