Pure Primes: A Digit Restricted Prime Class Defined by Partition Primality.

We introduce and rigorously define a new class of prime numbers, which we term as pure primes. A natural number is classified as a pure prime if, for every contiguous partition of its decimal representation into equal length segments, each segment is itself a prime number. Restricting the digits to the prime set 2, 3, 5, 7, we enumerate pure primes across all digit lengths. Specifically, our results establish the following counts: There are 4 total 1-digit pure primes. There are 4 total 2-digit pure primes. There are 15 total 3-digit pure primes. There is only 1 4-digit pure prime. There are 128 total 5-digit pure primes. There are 0 total 6-digit pure primes. There are 1325 total 7-digit pure primes. There are 0 total 8-digit pure primes. There are 469 total 9-digit pure primes. There are 0 total 10-digit pure primes. There are 214432 total 11-digit pure primes. There are 0 total 12-digit pure primes. There are 2884201 total 13-digit pure primes. There are 10 total 14-digit pure primes. There are 236 total 15-digit pure primes. As a result, the total pure primes across 1-15 digits are 3100825. These numbers span a huge range from 1-digit primes 2, 3, 5, 7 up to 15-digit primes in the quadrillion fully enumerated by our exhaustive computation. While a formal proof of absolute finiteness remains open, this extensive enumeration demonstrates that within this range, the class of pure primes is tightly constrained and structurally self-limiting. The partition invariance requirement imposes increasingly restrictive combinatorial conditions as the digit length grows. Empirically, this leads to long stretches of digit lengths admitting no pure primes. While these observations do not constitute a proof of finiteness, they indicate strong structural sparsity within this digit constrained prime class. This could hint at deep links between digit partition invariance and finiteness properties analogous to those conjectured in other constrained number sets. Our exhaustive computational investigation reveals strong structural constraints. Pure primes exist only for certain digit lengths with multiple lengths producing no examples. Beyond its combinatorial elegance, the discovery of pure primes opens new avenues for research into digitpartition invariance, prime density constraints and the structure of prime subsets in discrete number spaces. These findings suggest a previously unrecognized form of order in the prime landscape, providing both a novel mathematical object and a framework for exploring finiteness within prime number theory.