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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.

R_{35}^{(b)} : 11111111111111111111111111111111111 : 11111 * 1000010000100001000010000100001 : 1111111 * 10000001000000100000010000001,

Repunit is expressed in the base b.

In any positive multiples of Repunit, R_{n}^{(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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Daily Maths Topic Today - repunit fun.