The Product-Index Prime Conjecture: Corrected Reductions and Requirements for a Complete Computation

Abstract

Let p_n denote the nth prime and let D(m) be the product of the decimal digits of m. A product-index prime is a prime p_n satisfying D(p_n) = n. The known examples are p_7 = 17, p_21 = 73, and p_181440 = 2475989, and it has been conjectured that these are the only examples. This reviewed manuscript establishes the unconditional reductions that every product-index prime is smaller than 10^45 and that its index is 7-smooth. It also gives the exact size, 1,857,153, of the convenient superset of 7-smooth indices n ≤ 9^45. We then audit the proposed exhaustive-computation argument. Several defects in the earlier manuscript are mathematically consequential: the exponent of 5 was incorrectly set to zero, an explicit nth-prime bound was quoted with the wrong formula, the interval-to-prefix conversion was off by one, and the digit-completion recursion omitted the number of remaining positions. In addition, the supplied archive contains neither executable enumeration code nor the claimed certificates. We provide corrected interval and digit-dynamic-programming lemmas and specify the artifacts required for a reproducible proof. On the evidence supplied, the classification of all product-index primes remains a conjecture rather than an established theorem. The deposited archive contains the full LaTeX source, bibliography, compiled PDF, review notes, and an exact-integer Python script verifying the 1,857,153 count of 7-smooth indices.