105 (number)

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

105 is the 14th triangular number,^[1] a dodecagonal number,^[2] and the first Zeisel number.^[3] It is the first odd sphenic number and is the product of three consecutive prime numbers. 105 is the double factorial of 7.^[4] 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 numbers less than a thousand are 9, 15, 195, and 825.

105 is also the middle of the only prime sextuplet (97, 101, 103, 107, 109, 113) between the ones occurring at 7-23 and at 16057–16073. 105 is the product of the first three odd primes (${\displaystyle 3\times 5\times 7}$) and is less than the square of the next prime (11) by > 8. Therefore, for ${\displaystyle n=105}$, n ± 2, ± 4, and ± 8 must be prime (a prime k-tuple). In contrast, n ± 6, ± 10, ± 12, and ± 14 must be composite, making a prime gap on either side.

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 ${\displaystyle \pm 1}$.

105 is the number of parallelogram polyominoes with 7 cells.^[5]

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

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