Yagel Numbers: the Mersenne × Primorial Grid

The Mersenne × Primorial Grid

A Yagel number of order $k$ and exponent $n$ pushes the sieve idea behind Mersenne numbers ("exclude divisibility by 2") to its logical extreme:

$Y_k(n) \;=\; p_{k-1}\#\cdot p_k^{\,n} \;-\; 1 \;=\; 2\cdot 3\cdot 5\cdots p_{k-1}\cdot p_k^{\,n} - 1, \qquad k, n \ge 1$

where $p_k$ is the $k$ -th prime and $p_{k-1}\#$ the primorial of its predecessor. Both constants are dictated by the prime sequence itself — the family has no free parameters and is indexed purely by the lattice point $(k, n)$ . Two classical families appear as the boundary of the grid: row $k = 1$ consists of the Mersenne numbers $2^n - 1$ , and column $n = 1$ of the primorial numbers $p_k\# - 1$ . The companion face $Y^{+}_k(n) = p_{k-1}\#\cdot p_k^{\,n} + 1$ (the second kind) carries the Euclid and Fermat forms on its edges.

Three structural properties drive everything: $Y_k(n) \equiv -1 \pmod q$ for every prime $q \le p_k$ — the maximal built-in small-prime sieve for the base; no exponent is ever algebraically disqualified, so every lattice cell is a live candidate (unlike the Mersenne row); and $Y_k(n) + 1$ is fully factored by construction, so every prime in the family admits a fast deterministic certificate.

A Lucas–Lehmer-type test for every row

Primality of $Y_k(n)$ is decided by a single $V$ -only Lucas ladder — an $n$ -fold iteration of the degree- $p_k$ Chebyshev-type map $x \mapsto V_{p_k}(x)$ . For $k = 1$ the map is $x^2 - 2$ and the criterion collapses to the classical Lucas–Lehmer test verbatim. The cost is about two multiplications per bit, and the output is a proof, not a probability. The observed prime density across the grid matches the sieve-adjusted expectation exactly — no free parameters, in aggregate and row by row.

Running that machinery forward produced 1,079 certified Yagel primes — including $p_{999}\#\cdot 7919^{3918} + 1$ with 18,664 digits, a certified census of both faces for $k \le 50$ , a first certified prime in every row through $k = 100$ (milestones to $k = 1000$ ), and 22 twin pairs with both members proven. Explore any cell of the grid on the datasets page — it computes the exact value in your browser.

The full theory, statistics, and discovery history are presented in the living edition of the paper.