The Mersenne × Primorial Grid

Yagel numbers extend the sieve idea behind Mersenne numbers: exclude every prime through the base. Each prime base is coupled to the primorial of its predecessor, forming a canonical grid indexed by positive integers $k$ and $n$:

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

Here $p_k$ is the $k$-th prime and $p_{k-1}\mathrm{\#}$ is the product of the preceding primes, with $p_0\mathrm{\#}=1$. Row $k=1$ gives the Mersenne numbers $2^n-1$; column $n=1$ gives the primorial numbers $p_k\mathrm{\#}-1$. The companion face $Y_k^{+}(n)=p_{k-1}\mathrm{\#}\cdot p_k^n+1$ carries the Fermat and Euclid forms on its edges. Both coefficient and base are fixed by the prime sequence.

Explore the work

Why this grid

• A maximal built-in small-prime sieve. $Y_k(n)\equiv -1(\mathrm{mod}q)$ for every prime $q\le p_k$: no prime through the base divides a candidate.
• A live interior. For $k\ge 2$, no exponent is ruled out by an algebraic factorization in the exponent, unlike the Mersenne row. This does not establish infinitely many primes in a row.
• A factored neighbor at every cell. $Y_k(n)+1=p_{k-1}\mathrm{\#}\cdot p_k^n$ is completely factored by construction; on the companion face, $Y_k^{+}(n)-1$ is the same product. Every prime in the grid therefore admits a deterministic primality proof.

A proof for every prime in the grid

On the minus face, the classical Lucas $N+1$ criterion uses the factored neighbor; on the plus face, Pocklington's $N-1$ criterion does the same. A value enters the catalogue only with a successful deterministic proof for its exact integer.

In the size regime stated in the paper, and with a seed that succeeds, the minus-face criterion takes a single $V$-only Lucas ladder: an $n$-fold iteration of $x\mapsto V_{p_k}(x)$. The classical Lucas–Lehmer iteration $x\mapsto x^2-2$ is its $k=1$ case. These criteria are classical; the grid supplies the fully factored neighbor they require. The paper's primality section states the size, seed and coprimality hypotheses and the treatment of the exceptional Mersenne edge.

Current catalogue

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 $p_5\mathrm{\#}\cdot {13}^{44763}+1$, 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. Use Explore the Grid to compute any cell exactly in your browser and read its published verdict.

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