10

== Number ==
$10$ (ten) is:
:$2 \times 5$
:The base of the decimal system
:The only triangular number which is the sum of consecutive odd squares:
::$10 = 1^2 + 3^2$
:Probably the only number (except for the obvious $\paren {n!}! = n! \paren {n! - 1}!$ whose factorial is the product of $2$ factorials:
::$10! = 7! \, 6!$
:and so consequently:
::$10! = 7! \, 5! \, 3!$
:The $1$st power of $10$ after the zeroth $1$:
::$10 = 10^1$
:The $1$st noncototient:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 10$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $1$st positive integer with multiplicative persistence of $1$
:The $1$st non-square positive integer which cannot be expressed as the sum of a square and a prime
:The $2$nd positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $\ldots$
:The $2$nd number after $5$ to be the sum of two different squares:
::$10 = 1^2 + 3^2$
:The $2$nd after $1$ of the $5$ tetrahedral numbers which are also triangular
:The $2$nd positive integer after $1$ which is not the sum of a square and a prime:
::$10 = 1 + 9 = 4 + 6 = 9 + 1$: none of $1$, $6$ and $9$ are prime
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $2$ ways:
::$10 = 3 + 7 = 5 + 5$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $2$ different ways:
::$10 = 1 + 9 = 3 + 7$
:The $3$rd number after $2$ and $6$ that is not the difference of two squares, as it is of the form $4 n + 2$:
::$10 = 4 \times 2 + 2$
:The $3$rd tetrahedral number after $1$, $4$:
::$10 = 1 + 3 + 6 = \dfrac {3 \paren {3 + 1} \paren {3 + 2} } 6$
:The $3$rd happy number after $1$, $7$:
::$10 \to 1^2 + 0^2 = 1$
:The $3$rd after $2$, $5$ of $4$ numbers whose letters, when spelt in French, are in alphabetical order:
::dix
:The $4$th term of Göbel's sequence after $1$, $2$, $3$, $5$:
::$10 = \paren {1 + 1^2 + 2^2 + 3^2 + 5^2} / 4$
:The $4$th semiprime after $4$, $6$, $9$:
::$10 = 2 \times 5$
:The $4$th triangular number after $1$, $3$, $6$:
::$10 = 1 + 2 + 3 + 4 = \dfrac {4 \paren {4 + 1} } 2$
:The $4$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $\ldots$
:The $5$th even number after $2$, $4$, $6$, $8$ which cannot be expressed as the sum of $2$ composite odd numbers
:The $5$th after $1$, $2$, $5$, $6$ of $6$ integers $n$ such that the alternating group $A_n$ is ambivalent
:The $7$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$:
::$\map {\sigma_1} {10} = 18$
:The $7$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$ which cannot be expressed as the sum of distinct pentagonal numbers
:The $7$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime:
::$10! - 9! + 8! - 7! + 6! - 5! + 4! - 3! + 2! - 1! = 3 \, 301 \, 819$
:The $8$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$ which cannot be expressed as the sum of exactly $5$ non-zero squares
:The $9$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The $9$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
:The $10$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$:
::$10 = 10 \times 1 = 10 \times \paren {1 + 0}$
:The $10$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$ such that $5^n$ contains no zero in its decimal representation:
::$5^{10} = 9 \, 765 \, 625$
:The total of all the entries in a magic square of order $2$ (if it were to exist), after $1$:
::$10 = \ds \sum_{k \mathop = 1}^{2^2} k = \dfrac {2^2 \paren {2^2 + 1} } 2$
== Also see ==
* Divisibility by 10
* Divisibility by Power of 10
* 10 is Only Triangular Number that is Sum of Consecutive Odd Squares
* Factorial as Product of Two Factorials
* Factorial as Product of Three Factorials
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== Historical Note ==
== Linguistic Note ==
== Sources ==
* : Glossary
* : $10$
* : Glossary
* : $10$
Category:Specific Numbers
Category:10