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 1,073 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.