# 10

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## 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 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 $1$:
$10 = \ds \sum_{k \mathop = 1}^{2^2} k = \dfrac {2^2 \paren {2^2 + 1} } 2$

## Historical Note

Aristotle made the obvious statement that $10$ is the usual number base because we have $10$ fingers. This view was echoed by the Roman poet Ovid.

It is noted that not all cultures use $10$ -- some use $5$ (based on the number of fingers on a single hand) and some use $20$ (based on the total number of fingers and toes).

The number $10$ was considered holy by the Pythagoreans, who set great score to the fact that $10 = 1 + 2 + 3 + 4$.

They knew about $8$ celestial bodies:

Sun, Moon, Mercury, Venus, Earth, Mars, Jupiter, Saturn

and also believed in the Central Fire.

In order to satisfy their belief that $10$ was such a special number, they hypothesized the existence of a Counter-Earth which orbited the Central Fire in such a position as to be hidden from us by the Sun.

The number $10$ is expressed in Roman numerals as $\mathrm X$.

It has been suggested that this originates from:

joining up two $\mathrm V$s, each representing $2$ hands held up showing the full $5$ fingers

or:

an abbreviation for $10$ tally marks, with a line struck through them to indicate that $10$ had been reached.

## Linguistic Note

Words derived from or associated with the number $10$ include:

decade: a period of $10$ years
decimal: a system of counting in $10$s
decimate: to execute every $10$th man in line