In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit.
The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.
In fact, the base-2 repunits are the well-known Mersenne numbers Mn = 2n − 1, they start with (Prime factors colored red means "new factors", i. e. the prime factor divides Rn but does not divide Rk for all k < n) (sequence A102380 in the OEIS)[2] Smallest prime factor of Rn for n > 1 are The definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers.
cyclotomic polynomial and d ranges over the divisors of n. For p prime, which has the expected form of a repunit when x is substituted with b.
On April 3, 2007 Harvey Dubner (who also found R49081) announced that R109297 is a probable prime.
[3] On July 15, 2007, Maksym Voznyy announced R270343 to be probably prime.
[4] Serge Batalov and Ryan Propper found R5794777 and R8177207 to be probable primes on April 20 and May 8, 2021, respectively.
Particular properties are If b is a perfect power (can be written as mn, with m, n integers, n > 1) differs from 1, then there is at most one repunit in base-b.
It is also conjectured that when b is neither a perfect power nor −4k4 with k positive integer, then there are infinity many base-b repunit primes.
, the prime numbers will be distributed near the best fit line where limit
and there are about base-b repunit primes less than N. We also have the following 3 properties: Although they were not then known by that name, repunits in base-10 were studied by many mathematicians during the nineteenth century in an effort to work out and predict the cyclic patterns of repeating decimals.
[11] It was found very early on that for any prime p greater than 5, the period of the decimal expansion of 1/p is equal to the length of the smallest repunit number that is divisible by p. Tables of the period of reciprocal of primes up to 60,000 had been published by 1860 and permitted the factorization by such mathematicians as Reuschle of all repunits up to R16 and many larger ones.
By 1880, even R17 to R36 had been factored[11] and it is curious that, though Édouard Lucas showed no prime below three million had period nineteen, there was no attempt to test any repunit for primality until early in the twentieth century.
Further advances in the study of repunits did not occur until the 1960s, when computers allowed many new factors of repunits to be found and the gaps in earlier tables of prime periods corrected.
The Cunningham project endeavours to document the integer factorizations of (among other numbers) the repunits to base 2, 3, 5, 6, 7, 10, 11, and 12.
D. R. Kaprekar has defined Demlo numbers as concatenation of a left, middle and right part, where the left and right part must be of the same length (up to a possible leading zero to the left) and must add up to a repdigit number, and the middle part may contain any additional number of this repeated digit.
[13] They are named after Demlo railway station (now called Dombivili) 30 miles from Bombay on the then G.I.P.
He calls Wonderful Demlo numbers those of the form 1, 121, 12321, 1234321, ..., 12345678987654321.