109 (number)

← 108 109 110 →
← 100 101 102 103 104 105 106 107 108 109 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred nine
Ordinal	109th (one hundred ninth)
Factorization	prime
Prime	29th
Divisors	1, 109
Greek numeral	ΡΘ´
Roman numeral	CIX, cix
Binary	11011012
Ternary	110013
Senary	3016
Octal	1558
Duodecimal	9112
Hexadecimal	6D16

109 (one hundred [and] nine) is the natural number following 108 and preceding 110.

In mathematics

109 is the 29th prime number. As 29 is itself prime, 109 is the tenth super-prime. ^[1] The previous prime is 107, making them both twin primes. ^[2]

109 is a centered triangular number. ^[3]

There are exactly:

• 109 different families of subsets of a three-element set whose union includes all three elements. ^[4]
• 109 different loops (invertible but not necessarily associative binary operations with an identity) on six elements. ^[5]
• 109 squares on an infinite chessboard that can be reached by a knight within three moves. ^[6]

There are 109 uniform edge-colorings to the 11 regular and semiregular (or Archimedean) tilings. ^[7]

The decimal expansion of 1/109 can be computed using the alternating series, with ${\displaystyle F(n)}$ the ${\displaystyle n^{th}}$ Fibonacci number:

${\displaystyle {\frac {1}{109}}=\sum _{n=1}^{\infty }{F(n)\times 10^{-(n+1)}}\times (-1)^{n+1}=0.00917431\dots }$

The decimal expansion of 1/109 has 108 digits, making 109 a full reptend prime in decimal. The last six digits of the 108-digit cycle are 853211, the first six Fibonacci numbers in descending order. ^[8]

See also

• 109 (disambiguation)

References

1. Sloane, N. J. A. (ed.). "SequenceA006450(Primes with prime subscripts)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Sloane, N. J. A. (ed.). "SequenceA006512(Greater of twin primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
3. Sloane, N. J. A. (ed.). "SequenceA005448(Centered triangular numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
4. Sloane, N. J. A. (ed.). "SequenceA003465(Number of ways to cover an n-set)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
5. Sloane, N. J. A. (ed.). "SequenceA057771(Number of loops (quasigroups with an identity element) of order n)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
6. Sloane, N. J. A. (ed.). "SequenceA018836(Number of squares on infinite chess-board at ≤ n knight's moves from a fixed square)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
7. Asaro, Laura; Hyde, John; et al. (January 2015). "Uniform edge-c-colorings of the Archimedean tilings". Discrete Mathematics. 338 (1): 19–22. doi:10.1016/j.disc.2014.08.015. Zbl   1308.52017.
8. "89, 109, and the Fibonacci Sequence". May 15, 2012. Retrieved November 8, 2022.