105 (number)

105 (one hundred [and] five) is the natural number following 104 and preceding 106.

In mathematics[]

105 is a triangular number, a dodecagonal number^[1] and the first Zeisel number.^[2] It is a sphenic number, and is the product of three consecutive prime numbers. 105 is the double factorial of 7.^[3] It is also the sum of the first five square pyramidal numbers.

105 comes in the middle of the prime quadruplet (101, 103, 107, 109). The only other such odd numbers less than a thousand are 9, 15, 195 and 825. 105 is also a pseudoprime to the prime bases 13, 29, 41, 43, 71, 83 and 97. The distinct prime factors of 105 add up to 15, and so do those of 104, hence the two numbers form a Ruth-Aaron pair under the first definition.

105 is also a number n for which ${\displaystyle n-2^{k}}$ is prime, for ${\displaystyle 0<k<log_{2}(n)}$. (This even works up to ${\displaystyle k=8}$, ignoring the negative sign.)

105 is the smallest integer such that the factorization of ${\displaystyle x^{n}-1}$ over Q includes non-zero coefficients other than ${\displaystyle \pm 1}$. In other words, the 105th cyclotomic polynomial, Φ₁₀₅, is the first with coefficients other than ${\displaystyle \pm 1}$.

In science[]

• The atomic number of dubnium.

In other fields[]

105 is also:

• A Shimano Road groupset since 1984

See also[]

• List of highways numbered 105

References[]

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134