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 follow from that coupling. First, $Y_k(n) \equiv -1 \pmod q$ for every prime $q \le p_k$ : the maximal built-in small-prime sieve for the base. Second, for $k \ge 2$ no exponent is ever disqualified by an algebraic factorization, in contrast to the Mersenne row, where composite exponents cannot give primes. Third, $Y_k(n) + 1$ is completely factored by construction, so a deterministic primality proof is available at every cell of the grid. The second property concerns factorizations in the exponent only; it does not show that any row contains infinitely many primes.

A proof for every prime in the grid

Because one neighbor of $Y_k(n)$ is completely factored, the classical criteria apply at every cell: the Lucas $N+1$ test on the minus face, and Pocklington's $N-1$ test on the plus face. Each produces a proof rather than a probability, at roughly the cost of three probable-prime tests. Within the grid this takes a particularly compact form: in the size regime stated in the paper, and given a successful seed, primality is decided by a single $V$ -only Lucas ladder — an $n$ -fold iteration of the degree- $p_k$ Chebyshev-type map $x \mapsto V_{p_k}(x)$ , of which the classical Lucas–Lehmer iteration $x \mapsto x^2 - 2$ is the $k = 1$ case. Criteria of this kind for $K p^n \pm 1$ are classical; what the grid contributes is that the hypothesis they all need — a completely factored neighbor — is supplied at every cell by construction, so no cell is excluded for want of a factorization. The rest of each criterion's hypotheses still have to be met: the single ladder applies in the size regime stated in the paper and needs a seed that succeeds, and outside that regime a multi-factor argument is used. A value is published as prime only when the applicable criterion returns a certificate, never on the strength of a probable-prime test.

Applied across the grid, this has produced a catalogue of 5,751 proven Yagel primes — 2,889 on the minus face and 2,862 on the plus face. The largest is $p_{5}\#\cdot 13^{44763} + 1$ , of 49,867 decimal digits. 31 cells are prime on both faces at once, giving twin primes in which each member has a completely factored neighbor. Explore any cell of the grid on the datasets page, which computes the exact value in your browser.

The structure theory, the primality criteria and the computational results are set out in the living edition of the paper.