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.
$$ Y_k(n) = \left(\prod_{p=1}^{k-1} primes \right) \cdot p_k^n - 1, \quad n,k \in \mathbb{N}^* $$
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.
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.