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105 (one hundred ndfive) is the
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following 104 and preceding 106.


In mathematics

105 is the 14th
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
, a dodecagonal number, and the first Zeisel number. It is the first odd sphenic number and is the product of three consecutive
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s. 105 is the
double factorial In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same Parity (mathematics), parity (odd or even) as . That is, n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. Restated ...
of 7. 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 (3\times5\times7) and is less than the square of the next prime (11) by > 8. Therefore, for 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 A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-st and the ''n''-th prime numbers, i.e., :g_n = p_ - p_n. ...
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 n - 2^k is prime, for 0 < k < log_2(n). (This even works up to k = 8, ignoring the negative sign.) 105 is the smallest integer such that the factorization of x^n-1 over Q includes non-zero coefficients other than \pm 1. In other words, the 105th
cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th prim ...
, Φ105, is the first with coefficients other than \pm 1. 105 is the number of parallelogram polyominoes with 7 cells.


References

* Wells, D. '' The Penguin Dictionary of Curious and Interesting Numbers'' London: Penguin Group. (1987): 134 Integers {{Num-stub