What do you think about a deterministic generative grammar for the exponents of Mersenne primes.?

[Martin Doina]

The grammar expresses every Mersenne exponent \(p_n\) (for \(n \geq 5\)) as the product of two earlier exponents plus or minus the difference of two additional earlier exponents: \(p_n = p_a \times p_b \pm (p_c - p_d)\). We demonstrate that this formula, using only the largest product of earlier exponents less than the target, successfully generates 47 of the 48 analyzable exponents (M5 through M49, M51, M52). One exponent (M50) remains an exception that may require extension. All exponents can be generated by an additive Matryoshka recurrence \(M_n = \sum_{i < n} c_i M_i\) with small integer coefficients \(c_i \in \{1,2,3,4\}\). The nesting depth satisfies \(D(M_n) = n - 2\), forming a strictly linear chain from seeds {2, 3}. We identify a convergent seed ratio \(\varphi_M \approx 0.798148\) governing the asymptotic frequency of the seeds in full expansions. The grammar is supported by a binary ladder tree structure, modular fingerprint constraints, and a coefficient multiplier sequence based on \(M_{12} = 127\).

Full  repository and links:https://github.com/gatanegro/MERSENNE-COLLATZ

https://zenodo.org/records/19713989

thanks

[Allan Wechsler]

I'm not qualified to comment on the details of this criterion; I don't know enough number theory to know if an approach like this is possible. But I confess to some deep skepticism. I would like to see how well this "grammar" succeeds with random collections of exponents with approximately the same density.

My skepticism will decrease drastically when this approach yields its first new Mersenne exponent; I assume this has not happened yet, because if it had, the poster would have led with that fact.

-- Allan

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