Yagel Numbers

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:

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

where $p_k$ is the $k$-th prime and $p_{k-1}\mathrm{\#}$ 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 $p_k\mathrm{\#}-1$. The companion face $Y_k^{+}(n)=p_{k-1}\mathrm{\#}\cdot p_k^n+1$ (the second kind) carries the Euclid and Fermat forms on its edges.

Three structural properties drive everything: $Y_k(n)\equiv -1(\mathrm{mod}q)$ for every prime $q\le p_k$ — the maximal built-in small-prime sieve for the base; no exponent is ever algebraically disqualified in the interior (k ≥ 2); this does not prove infinitely many prime values; 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 $Y_k(n)$ has, in the stated size regime and with a successful seed, a certificate from 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 is compared with a first-order sieve heuristic on a historical window. This does not establish a Poisson law or a limiting row constant.

Running that machinery forward produced 5,756 certified Yagel primes — including $p_5\mathrm{\#}\cdot {13}^{44763}+1$ with 49,867 digits, a historical reported census (completion evidence is being audited) of both faces for $k\le 1000$ (every cell to 5,000 digits), and 31 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.