# 3

## Number

$3$ (**three**) is:

- The $2$nd prime number after $2$

- The only Fibonacci prime whose index is composite

### $1$st Term

- The $1$st odd prime

- The $1$st Proth prime:
- $3 = 1 \times 2^1 + 1$

- The smaller element of the $1$st pair of twin primes, with $5$

- The $1$st power of $3$ after the zeroth $1$:
- $3 = 3^1$

- The $1$st prime number of the form $4 n + 3$:
- $3 = 4 \times 0 + 3$

- The $1$st Mersenne number and Mersenne prime:
- $3 = 2^2 - 1$

- The $1$st Fermat number and Fermat prime:
- $3 = 2^{\paren {2^0} } + 1$

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

- The $1$st of an arithmetic sequence of primes:
- $3$, $5$, $7$

- The $1$st prime number $p$ which satisfies the equation $p^2 \divides \paren {10^p - 10}$:
- $3^2 \divides \paren {10^3 - 10}$

- The $1$st Euclid number after the zeroth $2$:
- $3 = p_1\# + 1 = 2 + 1$

- The $1$st prime number that can be found starting from the beginning of the decimal expansion of $\pi$ (pi):
- $3 (\cdotp 14159 \ldots)$

- The $1$st of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime

- The $1$st integer $n$ 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:
- $3! - 2! + 1! = 5$

- The $1$st prime number which is palindromic in both decimal and binary:
- $3_{10} = 11_2$

- The $1$st unique period prime: its period is $1$:
- $\dfrac 1 3 = 0 \cdotp \dot 3$

- The smallest positive integer the decimal expansion of whose reciprocal has a period of $1$:
- $\dfrac 1 3 = 0 \cdotp \dot 3$

- The $1$st integer $n$ for which the Ramanujan-Nagell equation $x^2 + 7 = 2^n$ has an integral solution:
- $1^2 + 7 = 8 = 2^3$

### $2$nd Term

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

- The $2$nd Sophie Germain prime after $2$:
- $2 \times 3 + 1 = 7$, which is prime

- The $2$nd triangular number after $1$:
- $3 = 1 + 2 = \dfrac {2 \paren {2 + 1} } 2$

- The index of the $2$nd Mersenne prime after $2$:
- $M_3 = 2^3 - 1 = 7$

- The $2$nd Fibonacci prime after $2$

- The $2$nd permutable prime after $2$

- The $2$nd Lucas number after $(2)$, $1$:
- $3 = 2 + 1$

- The $2$nd Lucas prime after $2$

- The $2$nd lucky number after $1$:
- $1$, $3$, $\ldots$

- The $2$nd palindromic lucky number after $1$:
- $1$, $3$, $\ldots$

- The $2$nd Stern number after $1$

- The $2$nd Stern prime after $1$

- The $2$nd Cullen number after $1$:
- $3 = 1 \times 2^1 + 1$

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

- The $2$nd (trivially) left-truncatable prime after $2$

- The $2$nd (trivially) right-truncatable prime after $2$

- The $2$nd (trivially) two-sided prime after $2$

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

- The $2$nd prime $p$ after $2$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
- $3 \# + 1 = 2 \times 3 + 1 = 7$

- The $2$nd after $2$ of the $4$ known primes $p$ such that $\dfrac {p^p - 1} {p - 1}$ is itself prime:
- $\dfrac {3^3 - 1} {3 - 1} = 13$

- The sum of the first $2$ factorials:
- $3 = 1! + 2!$

- The index (after $2$) of the $2$nd Woodall prime:
- $3 \times 2^3 - 1 = 23$

- The $2$nd number after $1$, and the only prime, whose divisor sum is square:

- $\map {\sigma_1} 3 = 4 = 2^2$

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

- The $2$nd positive integer after $1$ of which the product of its Euler $\phi$ function and its divisor count equals its divisor sum:
- $\map \phi 3 \map {\sigma_0} 3 = 2 \times 2 = 4 = \map {\sigma_1} 3$

- The $2$nd positive integer solution after $1$ to $\map \phi n = \map \phi {n + 1}$:
- $\map \phi 3 = 2 = \map \phi 4$

- The $2$nd positive integer which is not the sum of $1$ or more distinct squares:
- $2$, $3$, $\ldots$

- The $2$nd element of the Fermat set after $1$

- The $2$nd positive integer after $2$ which cannot be expressed as the sum of distinct pentagonal numbers

- The $2$nd integer $n$ after $1$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
- $\map {\sigma_0} 3 = 2$, $\map \phi 3 = 2$, $\map {\sigma_1} 3 = 4$

- The $2$nd of the $2$nd pair of consecutive integers whose product is a primorial:
- $2 \times 3 = 6 = 3 \#$

- The $2$nd prime number after $2$ to be of the form $n! + 1$ for integer $n$:
- $2! + 1 = 2 + 1 = 3$

- where $n!$ denotes $n$ factorial

- The $2$nd Lucas number after $1$ which is also triangular:
- $3 = \ds \sum_{k \mathop = 1}^2 k = \dfrac {2 \times \paren {2 + 1} } 2 = 2 + 1$

- The $2$nd Euclid prime after $2$:
- $3 = p_1\# + 1 = 2 + 1$

- The $2$nd odd positive integer after $1$ such that all smaller odd integers greater than $1$ which are coprime to it are prime

- The $2$nd odd positive integer after $1$ that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime

- The $2$nd minimal prime base $10$ after $2$

- The $2$nd positive integer solution after $1$ to the Ramanujan-Nagell equation $x^2 + 7 = 2^n$ for integral $n$:
- $3^2 + 7 = 16 = 2^4$

### $3$rd Term

- The $3$rd palindromic triangular number after $0$, $1$ whose index is itself palindromic:
- $3 = T_2$

- The $3$rd highly abundant number after $1$, $2$:
- $\map {\sigma_1} 3 = 4$

- The $3$rd Ulam number after $1$, $2$:
- $3 = 1 + 2$

- The $3$rd positive integer after $1$, $2$ such that all smaller positive integers coprime to it are prime

- The $3$rd integer after $0$, $1$ which is palindromic in both decimal and binary:
- $3_{10} = 11_2$

- The $3$rd (strictly) positive integer after $1$, $2$ which cannot be expressed as the sum of exactly $5$ non-zero squares

- The $3$rd after $0$, $1$ of the $5$ Fibonacci numbers which are also triangular

- The $3$rd palindromic triangular number after $0$, $1$

- The $3$rd of the trivial $1$-digit pluperfect digital invariants after $1$, $2$:
- $3^1 = 3$

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

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

- The $3$rd Ramanujan-Nagell number after $0$, $1$:
- $3 = 2^2 - 1 = \dfrac {2 \paren {2 + 1} } 2$

- The $3$rd (strictly) positive integer after $1$, $2$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

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

- The $3$rd after $0$, $2$ of the $6$ integers which are the middle term of a sequence of $5$ consecutive integers whose cubes add up to a square
- $1^3 + 2^3 + 3^3 + 4^3 + 5^3 = 225 = 15^2$

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

- The number of different representations of $1$ as the sum of $3$ unit fractions

- The number of integer partitions for $3$:
- $\map p 3 = 3$

### $4$th Term

- The $4$th Fibonacci number after $1$, $1$, $2$:
- $3 = 1 + 2$

- The $4$th integer $n$ after $0$, $1$, $2$ such that $2^n$ contains no zero in its decimal representation:
- $2^3 = 8$

- The $4$th integer $n$ after $0$, $1$, $2$ such that $5^n$ contains no zero in its decimal representation:
- $5^3 = 125$

- The $4$th integer $n$ after $0$, $1$, $2$ such that both $2^n$ and $5^n$ have no zeroes:
- $2^3 = 8$, $5^3 = 125$

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

- The $4$th integer $m$ after $0$, $1$, $2$ such that $m! + 1$ (its factorial plus $1$) is prime:
- $3! + 1 = 6 + 1 = 7$

- The $4$th palindromic integer after $0$, $1$, $2$ which is the index of a palindromic triangular number
- $T_3 = 6$

- The $4$th palindromic integer after $0$, $1$, $2$ whose square is also palindromic integer
- $3^2 = 9$

### Arithmetic Functions on $3$

\(\ds \map {\sigma_0} { 3 }\) | \(=\) | \(\ds 2\) | $\sigma_0$ of $3$ | |||||||||||

\(\ds \map \phi { 3 }\) | \(=\) | \(\ds 2\) | $\phi$ of $3$ | |||||||||||

\(\ds \map {\sigma_1} { 3 }\) | \(=\) | \(\ds 4\) | $\sigma_1$ of $3$ |

## Also see

- Trisecting the Angle
- Three Points Describe a Circle
- Three Regular Tessellations
- Integer is Sum of Three Triangular Numbers
- Integer as Sum of Three Squares
- Divisibility by 3
- Smallest Magic Square is of Order 3
- Prime Fibonacci Number has Prime Index except for 3
- Prime whose Divisor Sum is Square is 3

### Previous in Sequence: $1$

*Previous ... Next*: Lucas Number*Previous ... Next*: Palindromes 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*: Solutions of Ramanujan-Nagell Equation*Previous ... Next*: Triangular Number*Previous ... Next*: Palindromic Triangular Numbers*Previous ... Next*: Sequence of Powers of 3*Previous ... Next*: Sum of Sequence of Factorials*Previous ... Next*: Lucky Number*Previous ... Next*: Sequence of Palindromic Lucky Numbers*Previous ... Next*: Fermat Set*Previous ... Next*: Cullen Number*Previous ... Next*: Sequence of Smallest Numbers whose Reciprocal has Period n*Previous ... Next*: Integers whose Phi times Divisor Count equal Divisor Sum*Previous ... Next*: Representation of 1 as Sum of n Unit Fractions*Previous ... Next*: Consecutive Integers with Same Euler Phi Value*Previous ... Next*: Numbers such that Divisor Count divides Phi divides Divisor Sum*Previous ... Next*: Ramanujan-Nagell Number*Previous ... Next*: Stern Number*Previous ... Next*: Stern Prime*Previous ... Next*: Triangular Fibonacci Numbers*Previous ... Next*: Numbers whose Divisor Sum is Square*Previous ... Next*: Triangular Lucas Numbers

### Previous in Sequence: $2$

#### Next in Sequence: $4$

*Previous ... Next*: Pluperfect Digital Invariant*Previous ... Next*: Zuckerman Number*Previous ... Next*: Harshad Number*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Ulam Number*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares*Previous ... Next*: Integers such that all Coprime and Less are Prime*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*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1

#### Next in Sequence: $5$

*Previous ... Next*: Prime Number*Previous ... Next*: Palindromic Prime*Previous ... Next*: Sophie Germain Prime*Previous ... Next*: Fibonacci Number*Previous ... Next*: Fibonacci Prime*Previous ... Next*: Index of Mersenne Prime*Previous ... Next*: Mersenne Prime/Historical Note*Previous ... Next*: Left-Truncatable Prime*Previous ... Next*: Right-Truncatable Prime*Previous ... Next*: Two-Sided Prime*Previous ... Next*: Euler Lucky Number*Previous ... Next*: Göbel's Sequence*Previous ... Next*: Permutable Prime*Previous ... Next*: Sequence of Prime Primorial plus 1*Previous ... Next*: Consecutive Integers whose Product is Primorial*Previous ... Next*: Prime Numbers Composed of Strings of Consecutive Ascending Digits*Previous ... Next*: Integer Partition*Previous ... Next*: Minimal Prime

#### Next in Sequence: $6$

#### Next in Sequence: $7$ and above

*Previous ... Next*: Lucas Prime*Previous ... Next*: Euclid Number*Previous ... Next*: Euclid Prime*Previous ... Next*: Prime Numbers of form Factorial Plus 1

*Previous ... Next*: Sequence of Integers whose Factorial plus 1 is Prime*Previous ... Next*: Palindromic Indices of Palindromic Triangular Numbers*Previous ... Next*: Square of Small-Digit Palindromic Number is Palindromic

### Next in Sequence: $4$

*Next*: Sequence of Integers whose Factorial minus 1 is Prime*Next*: Sum of Sequence of Alternating Positive and Negative Factorials being Prime*Next*: 91 is Pseudoprime to 35 Bases less than 91*Next*: Solutions of Ramanujan-Nagell Equation

### Next in Sequence: $5$

*Next*: Proth Prime*Next*: Fermat Number*Next*: Fermat Prime*Next*: Twin Primes*Next*: Sequence of Prime Primorial minus 1*Next*: Palindromes in Base 10 and Base 2*Next*: Palindromic Primes in Base 10 and Base 2

### Next in Sequence: $6$

### Next in Sequence: $7$

*Next*: Mersenne Number*Next*: Mersenne Prime*Next*: Lucky Number*Next*: Sequence of Palindromic Lucky Numbers

### Next in Sequence: $8$ and above

*Next*: Unique Period Prime*Next*: Consecutive Integers with Same Euler Phi Value*Next*: Numbers whose Divisor Sum is Square*Next*: Prime Numbers Embedded in Digits of Pi*Next*: Squares equal to Sum of 2 Cubes*Next*: Solutions to p^2 Divides 10^p - 10

## Historical Note

The number **$3$** was considered by the ancient Greeks to be the first odd number, as they did not consider $1$ (one) a number, as such.

They associated the number **$3$** with the triangle, with its **$3$** vertices and **$3$** sides.

To the Pythagoreans, odd and even numbers were considered to be either male or female, but sources differ on which was which.

Some suggest that $3$ was considered to be the first male number, being composed of unity ($1$) and $2$, the principle of **diversity**.

Such sources state that in contrast, the even numbers were considered to be female.

However, other sources suggest that it was the odd numbers which were female, while the even numbers were male.

In addition to that, in the eyes of the Pythagoreans, **$3$** was in fact the first number, as in addition they considered that $2$ was not a number either, as it had a beginning and an end, but no middle.

Proclus similarly considered **$3$** to be the first number, but his reason was that it was the first number to be increased more by multiplication than by addition: $3 \times 3$ is greater than $3 + 3$.

**$3$** is a common number into which to divide a body into parts.

For example:

- The
**positive**,**comparative**and**superlative**of natural language.

- The world is divided into $3$ parts:
**the Underworld**,**the Earth**(or**Middle-Earth**), and**the Heavens**.

- In the English language, the sequence (beloved of fairy tales)
**once**,**twice**,**thrice**ends there -- there is no single word for "$n$ times" for any higher number.

In many cultures in history, **$3$** is particularly significant.

In Greek mythology, there were:

- $3$ Fates
- $3$ Furies
- $3$ Graces
- $3 \times 3$ Muses
- Paris had to choose between $3$ goddesses

Oaths are repeated $3$ times.

Saint Peter denied Christ $3$ times.

The Bellman states, in *The Hunting of the Snark*, that:

*What I tell you three times is true.*

## Linguistic Note

Words derived from or associated with the number $3$ include:

**tripod**: a stand with $3$ legs

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3$

- Lucas Numbers/Examples
- Triangular Numbers/Examples
- Powers of 3/Examples
- Lucky Numbers/Examples
- Cullen Numbers/Examples
- Ramanujan-Nagell Numbers/Examples
- Stern Numbers/Examples
- Stern Primes/Examples
- Numbers whose Divisor Sum is Square/Examples
- Pluperfect Digital Invariants/Examples
- Zuckerman Numbers/Examples
- Harshad Numbers/Examples
- Highly Abundant Numbers/Examples
- Ulam Numbers/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Prime Numbers/Examples
- Palindromic Primes/Examples
- Sophie Germain Primes/Examples
- Fibonacci Numbers/Examples
- Fibonacci Primes/Examples
- Indices of Mersenne Primes/Examples
- Mersenne's Assertion/Examples
- Left-Truncatable Primes/Examples
- Right-Truncatable Primes/Examples
- Two-Sided Primes/Examples
- Euler Lucky Numbers/Examples
- Göbel's Sequence/Examples
- Permutable Primes/Examples
- Integer Partitions/Examples
- Minimal Primes/Examples
- Woodall Primes/Examples
- Lucas Primes/Examples
- Euclid Numbers/Examples
- Euclid Primes/Examples
- Proth Primes/Examples
- Fermat Numbers/Examples
- Fermat Primes/Examples
- Twin Primes/Examples
- Mersenne Numbers/Examples
- Mersenne Primes/Examples
- Unique Period Primes/Examples
- Specific Numbers
- 3