# 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

### Previous in sequence: $1$

*Previous ... Next*: Numbers not Sum of Square and Prime*Previous ... Next*: Sum of Terms of Magic Square*Previous ... Next*: Sequence of Powers of 10*Previous ... Next*: Tetrahedral and Triangular Numbers

### Previous in sequence: $4$

### Previous in sequence: $5$

*Previous ... Next*: Göbel's Sequence*Previous ... Next*: Roman Numerals*Previous ... Next*: Letters of Names of Numbers in Alphabetical Order/French

### Previous in sequence: $6$

*Previous ... Next*: Alternating Groups that are Ambivalent*Previous ... Next*: Triangular Number*Previous ... Next*: Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways

### Previous in sequence: $7$

*Previous ... Next*: Happy Number

### Previous in sequence: $8$

*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum

### Previous in sequence: $9$

*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation

*Previous ... Next*: Harshad Number*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares*Previous ... Next*: 91 is Pseudoprime to 35 Bases less than 91*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1

*Previous ... Next*: Semiprime Number

### Next in sequence: $25$ and above

*Next*: Smallest Arguments for given Multiplicative Persistence*Next*: Noncototient*Next*: Non-Square Positive Integers not Sum of Square and Prime

## 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:

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

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $10$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $10$

- Powers of 10/Examples
- Tetrahedral Numbers/Examples
- Göbel's Sequence/Examples
- Triangular Numbers/Examples
- Happy Numbers/Examples
- Highly Abundant Numbers/Examples
- Harshad Numbers/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Semiprimes/Examples
- Multiplicative Persistence/Examples
- Noncototients/Examples
- Specific Numbers
- 10