Repunit

Repunit is a portmanteau of the English words repeated (repeated) and unit (unit) and refers to a number that contains only the digit 1. A repunit is a special repdigit ( " Schnapszahl "); the name repunit was coined in 1966 by Albert H. Beiler. In German, the term ones or ones column snake used.

A prime repunit or repunit prime number is a repunit, which is also a prime number.

Definition

Mathematically Repunits ( in decimal ) is defined as

The number Rn thus consists of n ones. The sequence of Repunits begins 1, 11, 111, 1111, ... ( sequence A002275 in OEIS ).

Repunit primes

The definition of Repunits arose historically in search of a decomposition of such numbers into their prime factors. The question of whether a repunit is prime, busy even serious mathematician in the 19th century. So Carl Gustav Jacob Jacobi wrote a work entitled investigate whether the number 11111111111 is a prime number or not. A curiosity, prompted by Dase.

It is easy to show that Rn is divisible by R, if n is divisible by a. For example, R9 is divisible by R3. 111111111 = 111 · 1001001 Therefore must of necessity be a prime number n, so that Rn can be a prime number. However, this condition is not sufficient, for example, R3 is not a prime number, as R3 = 111 = 3 * 37

Except for this example of R3 can only divisors of p Rn be ( for a prime n ) when p = 2kn 1 for a given k

Repunit primes are rare. Rn is prime for n = 2, 19, 23, 317, 1031, ... ( sequence A004023 in OEIS ). The September 1999 by Harvey Dubner and in October 2000 found by Lew Baxter R49081 and R86453 are likely primes. End of March 2007 determined Bourdelais Paul and Harvey Dubner R109297 as prime- suspect, four months later found Maksym Voznyy and Anton Budnyy R270343 as present ( 2013) the largest known probable repunit prime. It is believed that there are infinitely many primes repunit.

Generalized Repunits

Since the above definition of Repunits is based on the decimal system, this definition may seem arbitrary. However, it is the idea underlying generalize by Repunits respect to b defines an arbitrary basis:

It is easy to prove that for each n, which is not without remainder by 2 or divisible p, 2p exists a repunit to the base, which is a multiple of n.

The base 2 are known as the Repunits Mersenne numbers Mn = 2n - 1

The repunit primes are a subset of permutable primes, ie primes that remain prime when reversed their numbers arbitrarily.

A particularly large generalized repunit prime number with 37 090 points calculated Andy Steward 2006. In 2010, Tom Wu was with an even greater with 41 832 points.

Repunit prime number to different bases

The first repunit primes to the base 3 are

With the associated n of 3, 7, 13, 71, 103, ... ( in sequence A028491 OEIS ).

For base 4 exists only repunit prime 5 (), as and 3 for odd n is a divisor of and for even n is a divisor of.

The first repunit primes to the base 5 are

With the associated n of 3, 7, 11, 13, 47, ... ( in sequence A004061 OEIS ).

The first repunit primes to the base 6 are

With the associated n of 2, 3, 7, 29, 71, ... ( in sequence A004062 OEIS )

The first repunit primes to the base 7 are

With the associated n of 5, 13, 131, 149, ... ( in sequence A004063 OEIS )

At the base 8 exists only repunit prime 73 (), and since the first factor is divisible by 7 if n is not divisible by 3 and the second factor is divisible by 7 if n is a multiple of three. At the base 9 there is no repunit primes, and since both are straight.