105 (one hundred
ndfive) is the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
following
104 and preceding
106.
In mathematics
105 is a
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
A dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for ''n'' is given by the formula
:D_=5n^2 - 4n
The first few dodecagonal numbers are:
: 0, 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672, 7 ...
, and the first
Zeisel number A Zeisel number, named after Helmut Zeisel, is a square-free integer ''k'' with at least three prime factors which fall into the pattern
:p_x = ap_ + b
where ''a'' and ''b'' are some integer constants and ''x'' is the index number of each prime f ...
. It is the first odd
sphenic number
In number theory, a sphenic number (from grc, σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers.
Definit ...
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 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 because the only ways ...
s. 105 is the
double factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is,
:n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots.
For even , the ...
of 7. It is also the sum of the first five
square pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broa ...
s.
105 comes in the middle of the
prime quadruplet
In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.
Prim ...
(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
In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.
Prim ...
(97, 101, 103, 107, 109, 113) between the ones occurring at 7-23 and at 16057–16073. As the product of the first three odd primes (
) and less than the square of the next prime (11) by > 8, for
, n ± 2, ± 4, and ± 8 must be prime, and n ± 6, ± 10, ± 12, and ± 14 must be composite (
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)-th and the
''n''-th prime numbers, i.e.
:g_n = p_ - p_n.\
W ...
).
105 is also a
pseudoprime
A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy.
Some sources use the term pseudoprime to ...
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
is prime, for
. (This even works up to
, ignoring the negative sign.)
105 is the smallest integer such that the factorization of
over Q includes non-zero coefficients other than
. 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 primiti ...
, Φ
105, is the first with coefficients other than
.
105 is the number of parallelogram polyominoes with 7 cells.
In science
*The
atomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...
of
dubnium
Dubnium is a Synthetic element, synthetic chemical element with the Symbol (chemistry), symbol Db and atomic number 105. It is highly radioactive: the most stable known isotopes of dubnium, isotope, dubnium-268, has a half-life of about 16 ho ...
.
In other fields
105 is also:
* A
Shimano Road groupset since 1984
See also
*
List of highways numbered 105
Route 105 or Highway 105 may refer to:
Argentina
* National Route 105
* La Pampa Provincial Route 105
Canada
* New Brunswick Route 105
* Nova Scotia Highway 105 (Trans-Canada Highway)
* Ontario Highway 105
* Prince Edward Island Route 105
* ...
References
* Wells, D. ''
The Penguin Dictionary of Curious and Interesting Numbers
''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary number theory written by David Wells. The first edition was published in paperback by Penguin Books in 1986 in the UK, ...
'' London: Penguin Group. (1987): 134
{{DEFAULTSORT:105 (Number)
Integers