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Repunit fun

Repunit Fun

Repunit function was first coined by Albert H Beiler in the book, Recreations in the Theory of Numbers. It is a whole number that contains only the digit 1. It is a repeating series of the digit 1 like 1, 11, 111, 1111, 11111...

Repunit prime is also widely called as Mersenne primes (prime number which is one less than the power of two). Repunit numbers have some fun properties, for example

111 * 111 : 12321,

1111 * 1111 : 1234321,

11111 * 11111 : 123454321.

222 + (333)2 : 111111,

2222 + (3333)2 : 11111111,

22222 + (33333)2 : 1111111111.

Properties of Repunit Numbers:

Any repunit in any base having a composite number of digits is necessarily composite.

R35(b) : 11111111111111111111111111111111111 : 11111 * 1000010000100001000010000100001 : 1111111 * 10000001000000100000010000001,

Repunit is expressed in the base b.

In any positive multiples of Repunit, Rn(b) contains at least n nonzero digits in base b.

The only known numbers that are repunits with at least 3 digits in more than one base simultaneously are 31 (111 in base 5, 11111 in base 2) and 8191 (111 in base 90, 1111111111111 in base 2).

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