The Irrational-Decimal Prime-Block Conjecture

The Irrational-Decimal Prime-Block Conjecture asks do primes turn up in the digits of irrational numbers at the rate you'd expect from ordinary integers? The construction is what makes this different from prior "primes in π" searches. Rather than scanning arbitrary digit windows, it slices the decimal expansion of a constant like π, e, or √2 into non-overlapping, growing blocks — nine 1-digit blocks, then ninety 2-digit blocks, then nine hundred 3-digit blocks, and so on — so that each region mirrors the count of ordinary integers of that length (1–9, 10–99, 100–999, …). Every block is read as an integer and checked for primality. The conjecture: the fraction of prime blocks at length k should decay just as the Prime Number Theorem predicts for ordinary k-digit integers, roughly 1/(k ln 10), for any irrational assumed to be "random enough" (ideally base-10 normal). A normalized diagnostic, R_x(k) = P_x(k)·k·ln 10, is proposed to test this directly, with convergence to 1 as 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, back this up: prime-block density falls from ~16–18% at k=3 to ~9.5–9.8% at k=5, tracking the exact density of primes among ordinary integers of the same size within about half a point — and the seven constants converge toward statistically indistinguishable behavior as block length grows. The paper is careful about what this does and doesn't show. The pattern is consistent with a random-digit model, but base-10 normality — 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 particular, position-dependent sequence of variable-length blocks, which normality's qualitative, infinite-limit guarantee doesn't by itself supply. The note positions itself explicitly as a conjecture with numerical support, not a theorem, and situates the construction against existing literature on primes in π's digits (Byatt–Dalrymple–Turner, relevant OEIS sequences) and the broader rigorous work on digit-restricted primes (Maynard, 2019).