repdigit

A repdigit is a positive integer that, in a fixed base b, has n digits all equal to the same digit d, with 1 ≤ d ≤ b − 1 and n ≥ 1. Equivalently, it has the value N = d × (b^n − 1)/(b − 1), since (b^n − 1)/(b − 1) is the n-digit string of ones in base b.

In base 10, repdigits include the single-digit numbers 1 through 9, as well as multi-digit numbers such as 11, 22, 33, …, 99, and 111, 222, 333, etc. In base 2, the only allowed digit is 1, so repdigits are numbers written as a string of n ones: 1, 11_2 (=3), 111_2 (=7), 1111_2 (=15), and so on.

Primality properties follow from the factorization N = d × U_n, where U_n = (b^n − 1)/(b − 1). If n > 1 and d > 1, then N is composite because it equals the product of two integers greater than 1. The only possible primes with n > 1 occur when d = 1, i.e., repunits in base b. In decimal, this yields numbers consisting entirely of ones, such as 11, 1111111111111111111, and other repunit primes for certain lengths n. Known decimal repunit primes occur for specific values of n (for example, n = 2, 19, 23, 317, 1031 are among those producing primes). In general, whether a repunit is prime depends on the base and the length.

Repdigits are a simple class of palindromic numbers and are often discussed in number theory and recreational mathematics for their symmetry and base-dependent properties. See also repunit for the all-ones variant.