The Irrational-Decimal Prime-Block Conjecture

The Irrational-Decimal Prime-Block Conjecture: partition the decimal digits of an irrational number into non-overlapping blocks whose length grows to match the natural-number scale — nine 1-digit blocks, then ninety 2-digit blocks, then nine hundred 3-digit blocks, and so on, each region sized to match the count of ordinary integers of that length (1–9, 10–99, 100–999, …). Read every block as an integer and test it for primality. The conjecture is that the probability of a length-k block being prime, Pₓ(k), decays at the same rate the Prime Number Theorem predicts for ordinary k-digit integers — roughly 1/(k ln 10) — for irrationals "random enough" in their digits, ideally base-10 normal. This growing-block structure is what separates it from earlier "primes in π" searches, which typically scan arbitrary fixed-length digit windows rather than a length tied to the natural-number scale. A diagnostic, Rₓ(k) = Pₓ(k) / P₍exact₎(k) — the ratio of the observed probability to the exact probability of primality among ordinary k-digit integers of the same size — tests this directly. Convergence to 1 is the signature of the conjecture holding. Computational tests across seven constants (√2, √3, √5, π, e, φ, ln 2), out to 5-digit blocks using close to 490,000 digits per constant, support this: prime-block probability falls from about 16–18% at k=3 to about 9.5–9.8% at k=5, tracking the exact probability among ordinary integers of the same size to within about half a point, with all seven constants converging toward statistically indistinguishable behavior as block length grows. The property that would justify treating blocks as random integers — remains unproven for every constant tested, including π and e. Even granting normality, a proof would still need to control this specific, position-dependent sequence of variable-length blocks, which normality's qualitative, infinite-limit guarantee doesn't supply on its own. This is presented as a conjecture with numerical support, not a theorem, positioned against existing literature on primes in π's digits (Byatt, Dalrymple, and Turner; relevant OEIS sequences) and the broader rigorous work on digit-restricted primes (Maynard, 2019).