Repunitrelated

Repunitrelated is a descriptor used in number theory to refer to objects and properties tied to repunits—numbers whose base-b representation consists solely of the digit 1. In base 10, the n-digit repunit is R_n = (10^n - 1)/9, yielding the sequence 1, 11, 111, 1111, and so on. More generally, in base b, the n-digit repunit is R_n(b) = (b^n - 1)/(b - 1) = 1 + b + b^2 + ... + b^{n-1}.

Key properties and relations are often described as repunit-related. A fundamental arithmetic fact is that gcd(R_m, R_n) = R_gcd(m,n). Consequently, R_m divides R_n whenever m divides n. This divisibility structure extends across bases: in base 2, repunits take the form R_n(2) = 2^n - 1, which are the Mersenne numbers, while in base 10 they are the familiar decimal repunits.

Repunit primes form a notable subtopic in repunit-related study. A base-10 repunit prime is a number R_p with p prime such that R_p is prime. Known examples occur for p in {2, 19, 23, 317, 1031}; it is unknown whether infinitely many repunit primes exist. More generally, primality of R_n(b) in other bases b depends on both n and b and is a subject of computational and theoretical research.

The term repunitrelated is not a standard named category in every mathematical text, but it is used to describe concepts, sequences, or problems that arise from or interact with repunits.

See also: repunit, repunit prime, Mersenne numbers.