We introduce and study a new class of prime numbers which we call Reversible Square-Square Root Primes (RSSR primes). A prime is an RSSR prime if the integer obtained by replacing each decimal digit of with and concatenating the results is itself prime and furthermore the prime is recoverable from by replacing each decimal digit of with its integer square root and concatenating. The defining property is therefore not merely that a digit transformation preserves primality in one direction but that it does so bijectively. The transformation and its inverse both map primes to primes and together they close a complete cycle. A central theoretical observation is that the digit set is not a hypothesis imposed on the definition but a necessary consequence of demanding invertibility. Since the map must be injective on single digits and its values must themselves be single digits to preserve digit count, the admissible set is forced to be exactly whose squares are precisely the single-digit perfect squares. Any prime whose decimal representation contains a digit outside this set cannot satisfy the RSSR property. The constraint is structural, not computational and constitutes the first theorem of the paper. We establish that the RSSR family contains a natural and provably characterizable subclass which we call Self-Referential RSSR primes (SR-RSSR primes). These are RSSR primes for which , that is, primes that are fixed points of the digit squaring map. We prove that the SR-RSSR primes coincide exactly with the binary-digit primes, namely the primes whose decimal representations use only the digits and . The coincidence is not empirical but follows immediately from the idempotence of squaring on the set since and We conduct a rigorous and exhaustive computational search for RSSR primes up to (one hundred trillion) using a deterministic Miller-Rabin primality test with a complete set of witnesses sufficient for all integers below ensuring that no probabilistic error is possible in our results. The search proceeds by generating all numbers with digits in systematically and for each prime in this set applying the digit-squaring map and testing the image for primality. The search space contains candidate numbers of which are prime and of these we find that exactly are RSSR primes. The SR-RSSR subclass contains exactly members in the same range. Analysis of the distribution of RSSR primes by digit length reveals a robust and consistent growth pattern. The count of -digit RSSR primes grows by a factor converging to approximately per additional digit from digit length onward while the total candidate space grows by exactly a factor of per digit length. The ratio of RSSR primes to candidate primes decreases monotonically from digit length onward consistent with the prediction of the Prime Number Theorem applied to the restricted digit alphabet. The sequence of RSSR primes does not appear in the Online Encyclopedia of Integer Sequences and to the best of our knowledge this class has not been previously studied. The results established here establish the RSSR primes as a well-defined, infinite, computationally accessible and theoretically grounded family within the landscape of digit-manipulation prime families.
Christoper Muoki Mututu (Mon,) studied this question.