10

Definition
$10$ (Ten) is:


 * The $2$nd number after $5$ to be the sum of two different squares:
 * $10 = 1^2 + 3^2$


 * 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 $4$th triangular number:
 * $10 = 1 + 2 + 3 + 4$


 * The only triangular number which is the sum of consecutive odd squares:
 * $10 = 1^2 + 3^2$


 * The $3$rd tetrahedral number after $1$ and $4$:
 * $10 = 1 + 3 + 6$


 * Probably the only number (except for the obvious $\left({n!}\right)! = n! \left({n! - 1}\right)!$ whose factorial is the product of $2$ factorials:
 * $10! = 7! \, 6!$
 * and so consequently:
 * $10! = 7! \, 5! \, 3!$


 * The smallest noncototient:
 * $\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 10$
 * where $\phi \left({m}\right)$ denotes the Euler $\phi$ function


 * The second 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 with multiplicative persistence of $1$.


 * The base of the decimal system.

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