repunits

Repunits are positive integers that consist entirely of the digit 1 in a given base. The most familiar case is base 10, where the n-digit repunit is denoted R_n and equals 111...1 with n ones. In closed form, R_n(b) = (b^n - 1)/(b - 1) = 1 + b + b^2 + ... + b^{n-1}, where b is the base.

General properties: For any base b ≥ 2, if n is composite then R_n(b) is composite. If n = d·k with d > 1, then R_n(b) = R_d(b) × (1 + b^d + b^{2d} + ... + b^{(k-1)d}). Therefore, a necessary condition for R_n(b) to be prime is that n is prime. A repunit prime in base b is a prime of the form (b^n - 1)/(b - 1) with n > 1.

Special cases: In base 2, R_n(2) = 2^n - 1, the Mersenne numbers; many of these are prime for certain prime n (Mersenne primes). In base 10, the first few repunits are 11, 111, 1111, 11111, 111111, and so on. Among these, 11 is prime while 111 = 3 × 37, 1111 = 11 × 101, 11111 = 41 × 271, and 111111 = 3 × 7 × 11 × 13 × 37.

Connections and applications: Repunits relate to cyclotomic factorization, since 10^n − 1 factors into a product involving repunits. The study of repunit primes is a topic in primality testing and computational number theory, with repunit primes known only for certain large n values in various bases.