# 5

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## Number

$5$ (**five**) is:

- The $3$rd prime number.

- The only known odd untouchable number, and probably the only one.

### $1$st Term

- The $1$st power of $5$ after the zeroth $1$:
- $5 = 5^1$

- The $1$st Pythagorean prime, and thus, from Fermat's Two Squares Theorem, the sum of two squares uniquely:
- $5 = 4 \times 1 + 1 = 2^2 + 1^2$

- The $1$st prime number of the form $6 n - 1$:
- $5 = 6 \times 1 - 1$

- The $1$st prime number of the form $n! - 1$ for integer $n$:
- $5 = 3! - 1$

- where $n!$ denotes $n$ factorial

- The first prime number of the form $\ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$:
- $5 = 3! - 2! + 1!$

- The $1$st safe prime:
- $5 = 2 \times 2 + 1$

- The length of the hypotenuse of the smallest Pythagorean triangle:

- The lower and upper ends of the $1$st record-breaking gap between twin primes, which in this case is no gap at all:
- $5 - 5 = 0$

- The $1$st of the smallest sequence of both $4$ and $5$ prime numbers in arithmetic sequence:
- $5$, $11$, $17$, $23$
- $5$, $11$, $17$, $23$, $29$

- The $1$st Thabit number after the zeroth: $2$, and $2$nd Thabit prime:
- $5 = 3 \times 2^1 - 1$

- The $1$st primorial prime:
- $5 = p_2 \# - 1 = 3 \# - 1 = 2 \times 3 - 1$

- The $1$st positive integer $n$ such that no factorial of an integer can end with $n$ zeroes

- The $1$st balanced prime:
- $5 = \dfrac {3 + 7} 2$

- The $1$st Wilson prime:
- $5^2 \divides \paren {5 - 1}! + 1 = 25$

### $2$nd Term

- The $2$nd pentagonal number after $1$:
- $5 = 1 + 4 = \dfrac {2 \paren {2 \times 3 - 1} } 2$

- The $2$nd pentagonal number after $1$ which is also palindromic:
- $5 = 1 + 4 = \dfrac {2 \paren {2 \times 3 - 1} } 2$

- The $2$nd element of the $1$st pair of twin primes, with $3$
- Also the $1$st element of the $2$nd pair of twin primes, with $7$
- Hence the only element of $2$ pairs of twin primes:
- $\tuple {3, 5}$ and $\tuple {5, 7}$

- The $2$nd square pyramidal number after $1$:
- $5 = 1 + 4 = \dfrac {2 \paren {2 + 1} \paren {2 \times 2 + 1} } 6$

- The $2$nd pentatope number after $1$:
- $5 = 1 + 4 = \dfrac {2 \paren {2 + 1} \paren {2 + 2} \paren {2 + 3} } {24}$

- The $2$nd automorphic number after $1$:
- $5^2 = 2 \mathbf 5$

- The $2$nd Proth prime after $3$:
- $5 = 1 \times 2^2 + 1$

- The $2$nd Fermat number, and thus Fermat prime, after $3$:
- $5 = 2^{\paren {2^1} } + 1$

- The $2$nd untouchable number after $2$

- The $2$nd term of the $3$-Göbel sequence after $1$, $2$:
- $5 = \paren {1 + 1^3 + 2^3} / 2$

- The $2$nd $n$ after $4$ such that $n! + 1$ is square: see Brocard's Problem

- The $2$nd prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime, after $3$:
- $5 \# - 1 = 2 \times 3 \times 5 - 1 = 29$

- The $2$nd number such that $2 n^2 - 1$ is square, after $1$:
- $2 \times 5^2 - 1 = 2 \times 25 - 1 = 49 = 7^2$

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- The $2$nd prime number after $2$ which divides the sum of all smaller primes:
- $1 \times 5 = 5 = 2 + 3$

- The $2$nd Sierpiński number of the first kind after $2$:
- $5 = 2^2 + 1$

- The $2$nd prime Sierpiński number of the first kind after $2$:
- $5 = 2^2 + 1$

- The $2$nd prime number of the form $n^2 + 1$ after $2$:
- $5 = 2^2 + 1$

- The $2$nd integer after $2$ at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied

- The $2$nd tri-automorphic number after $2$:
- $5^2 \times 3 = 7 \mathbf 5$

- The $2$nd after $2$ of $4$ integers whose letters, when spelt in French, are in alphabetical order:
**cinq**

- The $2$nd prime number after $3$ which is palindromic in both decimal and binary:
- $5_{10} = 101_2$

- The $2$nd of $6$ integers after $2$ which cannot be expressed as the sum of distinct triangular numbers

- The $2$nd odd number after $1$ which cannot be expressed as the sum of an integer power and a prime number

### $3$rd Term

- The $3$rd (trivial, $1$-digit, after $2$, $3$) palindromic prime

- The $3$rd Sophie Germain prime after $2$, $3$:
- $2 \times 5 + 1 = 11$, which is prime

- The index of the $3$rd Mersenne prime after $2$, $3$:
- $M_5 = 2^5 - 1 = 31$

- The $3$rd generalized pentagonal number after $1$, $2$:
- $5 = \dfrac {2 \paren {3 \times 2 - 1} } 2$

- The $3$rd trimorphic number after $1$, $4$:
- $5^3 = 12 \mathbf 5$

- The $3$rd Catalan number after $(1)$, $1$, $2$:
- $5 = \dfrac 1 {3 + 1} \dbinom {2 \times 3} 3 = \dfrac 1 4 \times 20$

- The $3$rd after $1$, $2$ of $6$ integers $n$ such that the alternating group $A_n$ is ambivalent

- The $3$rd term of Göbel's sequence after $1$, $2$, $3$:
- $5 = \paren {1 + 1^2 + 2^2 + 3^2} / 3$

- The $3$rd Fibonacci prime after $2$, $3$

- The $3$rd and last Fibonacci number after $0$, $1$ which equals its index

- The $3$rd permutable prime after $2$, $3$

- The $3$rd prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime, after $2$, $3$:
- $5 \# + 1 = 2 \times 3 \times 5 + 1 = 31$

- The $3$rd of the lucky numbers of Euler after $2$, $3$:
- $n^2 + n + 5$ is prime for $0 \le n < 4$

- The $1$st element of the $3$rd pair of consecutive integers whose product is a primorial:
- $5 \times 6 = 30 = 5 \#$

- The $3$rd Bell number after $(1)$, $1$, $2$

- The $3$rd (trivially) two-sided prime after $2$, $3$

- The $3$rd prime number after $2$, $3$ consisting (trivially) of a string of consecutive ascending digits

- The $3$rd non-negative integer $n$ after $0$, $1$ such that the Fibonacci number $F_n$ ends in $n$

- The $3$rd integer $n$ after $3$, $4$ 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:
- $5! - 4! + 3! - 2! + 1! = 101$

- The $3$rd odd positive integer after $1$, $3$ such that all smaller odd integers greater than $1$ which are coprime to it are prime

- The $3$rd odd positive integer after $1$, $3$ that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime

### $4$th Term

- The $4$th integer after $0$, $1$, $3$ which is palindromic in both decimal and binary:
- $5_{10} = 101_2$

- The $4$th after $1$, $2$, $4$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

- The index of the $4$th Mersenne number after $1$, $2$, $3$ which Marin Mersenne asserted to be prime

- The number of distinct free tetrominoes

### $5$th Term

- The $5$th Fibonacci number after $1$, $1$, $2$, $3$:
- $5 = 2 + 3$

- The $5$th of the (trivial $1$-digit) pluperfect digital invariants after $1$, $2$, $3$, $4$:
- $5^1 = 5$

- The $5$th of the (trivial $1$-digit) Zuckerman numbers after $1$, $2$, $3$, $4$:
- $5 = 1 \times 5$

- The $5$th of the (trivial $1$-digit) harshad numbers after $1$, $2$, $3$, $4$:
- $5 = 1 \times 5$

- The $5$th after $1$, $2$, $3$, $4$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

### $6$th Term

- The $6$th integer $n$ after $0$, $1$, $2$, $3$, $4$ such that $2^n$ contains no zero in its decimal representation:
- $2^5 = 32$

- The $6$th integer $n$ after $0$, $1$, $2$, $3$, $4$ such that $5^n$ contains no zero in its decimal representation:
- $5^5 = 3125$

- The $6$th integer $n$ after $0$, $1$, $2$, $3$, $4$ such that both $2^n$ and $5^n$ have no zero digits:
- $2^5 = 32$, $5^5 = 3125$

- The $6$th integer after $0$, $1$, $2$, $3$, $4$ which is (trivially) the sum of the increasing powers of its digits taken in order:
- $5^1 = 5$

### Miscellaneous

- The magic constant of a magic square of order $2$ (if it were to exist), after $1$:
- $5 = \ds \dfrac 1 2 \sum_{k \mathop = 1}^{2^2} k = \dfrac {2 \paren {2^2 + 1} } 2$

## Also see

- Fibonacci Numbers which equal their Index
- Odd Untouchable Numbers
- Volume of Unit Hypersphere
- Lamé's Theorem
- Abel-Ruffini Theorem
- Number is Sum of Five Cubes
- Conic Section through Five Points
- Five Platonic Solids
- Brocard's Problem
- Five Color Theorem

### Previous in sequence: $1$

*Previous*: Fibonacci Numbers which equal their Index*Previous ... Next*: Automorphic Number*Previous ... Next*: Pentagonal Number*Previous ... Next*: Square Pyramidal Number*Previous ... Next*: Magic Constant of Magic Square*Previous ... Next*: Pentatope Number*Previous ... Next*: Sequence of Palindromic Pentagonal Numbers*Previous ... Next*: Sequence of Powers of 5*Previous ... Next*: Sequence of Fibonacci Numbers ending in Index*Previous ... Next*: Odd Numbers not Sum of Prime and Power

### Previous in sequence: $2$

*Previous ... Next*: Thabit Number*Previous ... Next*: Thabit Prime*Previous ... Next*: Number of Free Polyominoes*Previous ... Next*: Catalan Number*Previous ... Next*: Bell Number*Previous ... Next*: Primes of Form n^2 + 1*Previous ... Next*: Prime Number Race between 4n+1 and 4n-1*Previous ... Next*: Sierpiński Number of the First Kind*Previous ... Next*: 3-Göbel Sequence*Previous ... Next*: Untouchable Number*Previous ... Next*: Prime Numbers which Divide Sum of All Lesser Primes*Previous ... Next*: Prime Sierpiński Numbers of the First Kind

### Previous in sequence: $3$

*Previous ... Next*: Prime Number*Previous ... Next*: Palindromic Prime*Previous ... Next*: Twin Primes*Previous ... Next*: Permutable Prime*Previous ... Next*: Two-Sided Prime*Previous ... Next*: Index of Mersenne Prime*Previous ... Next*: Mersenne Prime/Historical Note*Previous ... Next*: Prime Numbers Composed of Strings of Consecutive Ascending Digits*Previous ... Next*: Sequence of Prime Primorial plus 1*Previous ... Next*: Palindromes in Base 10 and Base 2*Previous ... Next*: Palindromic Primes in Base 10 and Base 2*Previous ... Next*: Odd Integers whose Smaller Odd Coprimes are Prime*Previous ... Next*: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares

*Previous ... Next*: Fibonacci Number

*Previous ... Next*: Göbel's Sequence

*Previous ... Next*: Sophie Germain Prime*Previous ... Next*: Euler Lucky Number*Previous ... Next*: Sequence of Prime Primorial minus 1

*Previous ... Next*: Proth Prime*Previous ... Next*: Fibonacci Prime

*Previous ... Next*: Fermat Number*Previous ... Next*: Fermat Prime

### Previous in sequence: $4$

*Previous ... Next*: Trimorphic Number*Previous ... Next*: Pluperfect Digital Invariant*Previous ... Next*: Zuckerman Number*Previous ... Next*: Harshad Number*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation*Previous ... Next*: Powers of 2 and 5 without Zeroes*Previous ... Next*: Numbers which are Sum of Increasing Powers of Digits*Previous ... Next*: Sum of Sequence of Alternating Positive and Negative Factorials being Prime*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum

### Next in sequence: $6$ and above

*Next*: Safe Prime*Next*: Record Gaps between Twin Primes*Next*: Roman Numerals*Next*: Numbers of Zeroes that Factorial does not end with*Next*: Pythagorean Prime*Next*: Wilson Prime*Next*: Sum of Sequence of Alternating Positive and Negative Factorials being Prime*Next*: Prime Numbers of form Factorial Minus 1*Next*: Primorial Prime*Next*: Balanced Prime

## Historical Note

The number **$5$** was the number associated by the Pythagoreans with **marriage**, the sum of $2$, the first female number, and $3$, the first male number.

According to Plutarch, the Pythagoreans also associated $5$ with **nature**, on account of its automorphic property.

The number $5$ is expressed in Roman numerals as $\mathrm V$.

It has been suggested that this originates from the shape of one hand held up showing the full $5$ fingers.

## Linguistic Note

Words derived from or associated with the number $5$ include:

**pentathlon**: a competition comprising $5$ athletic events

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$

Categories:

- Work To Do
- Automorphic Numbers/Examples
- Pentagonal Numbers/Examples
- Pyramidal Numbers/Examples
- Pentatope Numbers/Examples
- Powers of 5/Examples
- Generalized Pentagonal Numbers/Examples
- Tri-Automorphic Numbers/Examples
- Thabit Numbers/Examples
- Thabit Primes/Examples
- Catalan Numbers/Examples
- Bell Numbers/Examples
- Prime Number Races/Examples
- Sierpiński Numbers of the First Kind/Examples
- Göbel's Sequence/Examples
- Untouchable Numbers/Examples
- Prime Numbers/Examples
- Palindromic Primes/Examples
- Twin Primes/Examples
- Permutable Primes/Examples
- Two-Sided Primes/Examples
- Indices of Mersenne Primes/Examples
- Mersenne's Assertion/Examples
- Fibonacci Numbers/Examples
- Sophie Germain Primes/Examples
- Euler Lucky Numbers/Examples
- Proth Primes/Examples
- Fibonacci Primes/Examples
- Fermat Numbers/Examples
- Fermat Primes/Examples
- Trimorphic Numbers/Examples
- Pluperfect Digital Invariants/Examples
- Zuckerman Numbers/Examples
- Harshad Numbers/Examples
- Safe Primes/Examples
- Wilson Primes/Examples
- Primorial Primes/Examples
- Balanced Primes/Examples
- Specific Numbers
- 5