109 (number)

Summary

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

← 108 109 110 →
Cardinalone hundred nine
Ordinal109th
(one hundred ninth)
Factorizationprime
Prime29th
Divisors1, 109
Greek numeralΡΘ´
Roman numeralCIX
Binary11011012
Ternary110013
Senary3016
Octal1558
Duodecimal9112
Hexadecimal6D16

In mathematics edit

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   the   Fibonacci number:

 

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 edit

References edit

  1. ^ Sloane, N. J. A. (ed.). "Sequence A006450 (Primes with prime subscripts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A003465 (Number of ways to cover an n-set)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. ^ 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.
  6. ^ 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.
  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.