# 3

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## 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 integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
$3! - 1 = 6 - 1 = 5$

The $1$st of $3$ primes of the form $2 x^2 + 3$:
$2 \times 0^2 + 3 = 3$ (Next)

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

The $1$st integer whose square is the sum of $2$ coprime cubes:
$3^2 = 2^3 + 1^3$

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

### $2$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 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) 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 $1$ 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

### $3$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

### $4$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 $3$

## 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