Repunit Totally Explained

Everything about Repunit totally explained

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by A.H. Beiler. A repunit prime is a repunit that's also a prime number.

History

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 recurring decimals.
   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's 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 had been factored and it's curious that, though Edouard 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. The American mathematician Oscar Hoppe proved R₁₉ to be prime in 1916 and Lehmer and Kraitchik independently found R₂₃ to be prime in 1929.
   Further advances in the study of repunits didn't 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. R₃₁₇ was found to be a probable prime circa 1966 and was proved prime eleven years later, when R₁₀₃₁ was shown to be the only further possible prime repunit with fewer than ten thousand digits. It was proven prime in 1986, but searches for further prime repunits in the following decade consistently failed. However, there was a major side-development in the field of generalized repunits, which produced a large number of new primes and probable primes.
   Since 1999, four further probably prime repunits have been found, but it's unlikely that any of them will be proven prime in the foreseeable future because of their huge size.

Definition

The repunits are defined mathematically as ť

R_n=nge1.

In fact, the base-2 repunits are the well-respected Mersenne numbers M_n = 2ⁿ − 1. 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.
Example 1) the first few base-3 repunit primes are 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (sequence in OEIS), corresponding to

n

of 3, 7, 13, 71, 103 (sequence in OEIS).
Example 2) the only base-4 repunit prime is 5 (

11_4

), because

4^n-1=left(2^n+1
ight)left(2^n-1
ight)

, and 3 divides one of these, leaving the other as a factor of the repunit.
It is easy to prove

that given n, such that n isn't exactly divisible by 2 or p, there exists a repunit in base 2p that's a multiple of n.

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