For each digit i ∈ 0, …, 9, we consider the partition of the set of primes P into two complementary classes: the class of primes whose decimal representation does not contain the digit i, denoted Pi-, and the class of those that contain it at least once, denoted Pi+. We will first establish that every set Pi- has zero density in P, which ensures that its complement Pi+ is infinite in P. This property will then be generalized to any set of s digits, where 1 ≤ s ≤ 9. Building on this result and on J. Maynard’s 2019 theorem (2), which guarantees the infinity of Pi-, we will prove that the sets Pi+ contain arbitrarily long sequences of consecutive primes. We then present the results of a systematic numerical exploration up to 1011, which lead us to a conjecture regarding the boundedness of sequences of consecutive primes in Pi-. We observe a clear separation of the maximum lengths of these sequences into two families, depending on whether the digit i can (family A) or cannot (family B) appear in the last position of a prime number. The record for length is held by a sequence of 117 consecutive primes excluding the digit 4 (family B), while the shortest is a sequence of 37 consecutive primes excluding the digit 3 (family A). For any i, at least in the interval 2, 1011, we note the completely different behavior between the maximal sequences of Pi+ and the maximal sequences of Pi-, these two complementary sets being disjoint and infinite sets.
René-Louis Clerc (Sat,) studied this question.