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
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 a 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] and 109 squares on an infinite chessboard that can be reached by a knight within three moves. ^[6]

The decimal expansion of 1/109 has 108 digits, making 109 a full reptend prime. The last six digits of the 108-digit cycle are 853211, the first six Fibonacci numbers in descending order. ^[7]

The decimal expansion of 1/109 can also be computed using the alternating series

${\displaystyle {\frac {1}{109}}=\sum _{n=1}^{\infty }{F(n)\times 10^{-(n+1)}}\times (-1)^{n+1}=0.00917431\dots \ ,F(n)}$ being the n^th Fibonacci number.

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. "89, 109, and the Fibonacci Sequence". May 15, 2012. Retrieved November 8, 2022.