109 (number)

In mathematics

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

109 is a centered triangular number.

There are exactly:

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

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

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

::\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.

References

References

1. {{Cite OEIS. A006450
2. {{Cite OEIS. A006512
3. {{Cite OEIS. A005448
4. {{Cite OEIS. A003465
5. {{Cite OEIS. A057771
6. {{Cite OEIS. A018836
7. (January 2015). "Uniform edge-c-colorings of the Archimedean tilings". Discrete Mathematics.
8. (May 15, 2012). "89, 109, and the Fibonacci Sequence".