From Wikipedia, the free encyclopedia
Natural number
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
F
(
n
)
{\displaystyle F(n)}
the
n
t
h
{\displaystyle n^{th}}
Fibonacci number:
1
109
=
∑
n
=
1
∞
F
(
n
)
×
10
−
(
n
+
1
)
×
(
−
1
)
n
+
1
=
0.00917431
…
{\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
References
^
Sloane, N. J. A. (ed.).
"Sequence A006450 (Primes with prime subscripts)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A006512 (Greater of twin primes)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A005448 (Centered triangular numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A003465 (Number of ways to cover an n -set)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A057771 (Number of loops (quasigroups with an identity element) of order n )" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A018836 (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.
^ 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 .
^
"89, 109, and the Fibonacci Sequence" . May 15, 2012. Retrieved November 8, 2022 .
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000