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Mahadee–Kamrujjaman Conjecture on uniform digit distribution in large primes

Prove the Mahadee–Kamrujjaman Conjecture asserting that, in base 10, the digits of prime numbers are uniformly distributed as the magnitude of primes grows without bound; specifically, for each digit d in {0,1,2,3,4,5,6,7,8,9}, if P_n(d) denotes the probability (or relative frequency) of digit d occurring among primes less than 10^n, then lim_{n→∞} P_n(d) = 0.1.

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Background

The conjecture is motivated by experiments performed to design mutation operators for genetic algorithms that use decimal chromosome representations when factoring large semiprimes. The authors empirically examined digit frequencies in the base-10 representations of primes up to 109 and observed that the empirical probabilities for each digit approach 0.1 as the bound increases.

They formalize this observation as the Mahadee–Kamrujjaman Conjecture. Due to computational limits, their empirical validation is restricted to primes below 109, leaving a proof or disproof open. Establishing the conjecture would clarify whether base-10 digit frequencies in primes exhibit asymptotic uniformity and would have implications for heuristic design in cryptanalytic genetic algorithms.

References

Finally, based on our experimentation on the probability distribution of each digit in the construct of a big prime numbers to device efficient mutation operation for decimal chromosome representation in GA, we propose a novel conjecture on the distribution of digits on prime which is stated below: Digits of a prime are uniformly distributed as the length of primes tends to infinity. i.e. Let P_n (d) be the probability of occurrence of a digit, d in {0,1,2,…,9} in primes of length less than 10n then for any choice of d, lim_{n→∞} P_n(d)=0.1.

— Cryptanalysis of RSA Cryptosystem: Prime Factorization using Genetic Algorithm (2407.05944 - Mobin et al., 24 Jun 2024) in Section 2 (Theoretical Background), Conjecture (Mahadee-Kamrujjaman Conjecture)