Number Gossip

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Unique Properties of 8191

8191 is the smallest Mersenne prime p such that the Mersenne number M(p) = 2^p - 1 is composite

Rare Properties of 8191

A number of the form 2^p - 1 is called a Mersenne number if p is prime.

It was believed many years ago, that all Mersenne numbers are prime. This is not so, thus there is a separate entry for Mersenne prime numbers.

...,
31,
127,
2047,
8191,
131071,
524287,
8388607,
...

Mersenne prime

A Mersenne number which is also prime is called a Mersenne prime.

The drive to find prime numbers among Mersenne numbers supplies humanity with the largest known prime numbers.

...,
7,
31,
127,
8191,
131071,
524287,
2147483647,
...

Common Properties of 8191

Apocalyptic power

The number n is called an apocalyptic power if 2ⁿ contains the consecutive digits 666 (in decimal).

...,
8187,
8188,
8190,
8191,
8192,
8193,
8195,
...

Deficient

The number n is deficient if the sum of all its positive divisors except itself is less than n.

Compare with perfect and abundant numbers.

...,
8187,
8188,
8189,
8191,
8192,
8193,
8194,
...

Lucky

To build the lucky number sequence, start with natural numbers. Delete every second number, leaving 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ... . The second number remaining is 3, so delete every third number, leaving 1, 3, 7, 9, 13, 15, 19, 21, ... . The next number remaining is 7, so delete every 7th number, leaving 1, 3, 7, 9, 13, 15, 21, ... . The next number remaining is 9, so delete every ninth number, etc.

Those numbers were lucky they weren't crossed out.

...,
8169,
8173,
8175,
8191,
8193,
8221,
8223,
...

Odd

A number is odd if it is not divisible by 2.

Numbers that are not odd are even. Compare with another pair -- evil and odious numbers.

...,
8185,
8187,
8189,
8191,
8193,
8195,
8197,
...

Odious

The number n is odious if it has an odd number of 1's in its binary expansion.

Guess what evil numbers are.

...,
8185,
8186,
8188,
8191,
8192,
8195,
8197,
...

Prime

A prime is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.

Prime numbers are opposite to composite numbers.

...,
8167,
8171,
8179,
8191,
8209,
8219,
8221,
...

Square-free

A number is said to be square-free if its prime decomposition contains no repeated factors.

...,
8186,
8187,
8189,
8191,
8193,
8194,
8195,
...

Tanya Khovanova

Number Gossip

Unique Properties of 8191

Rare Properties of 8191

Common Properties of 8191