2

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Number

$2$ (two) is:

The $1$st prime number


The successor integer to the number $1$ (one).


The only even prime number


Zeroth Term

The $0$th (zeroth) Lucas number


The $0$th Euclid number:
$2 = p_0\# + 1 = 1 + 1$


The $0$th Thabit number, and $1$st Thabit prime:
$2 = 3 \times 2^0 - 1$


$1$st Term

The $1$st (strictly) positive even number


The $1$st (trivial, $1$-digit) palindromic prime


The $1$st Sophie Germain prime:
$2 \times 2 + 1 = 5$, which is prime


The index of the $1$st repunit prime:
$R_2 = 11$


The index of the $1$st Mersenne prime:
$M_2 = 2^2 - 1 = 3$


The index of the $1$st Woodall prime:
$2 \times 2^2 - 1 = 7$


The $1$st second pentagonal number:
$2 = \dfrac {1 \paren {3 \times 1 + 1} } 2$


The $1$st power of $2$ after the zeroth $1$:
$2^1 = 2$


The $1$st Fibonacci prime


The $1$st Lucas prime


The $1$st permutable prime


The $1$st untouchable number


The $1$st primorial:
$2 = p_1 \# = 2 \# := \displaystyle \prod_{k \mathop = 1}^1 p_k = 2$


The $1$st central binomial coefficient:
$2 = \dbinom {2 \times 1} 1 := \dfrac {2!} {\paren {1!}^2}$


The $1$st prime number of the form $n! + 1$ for integer $n$:
$1! + 1 = 1 + 1 = 2$
where $n!$ denotes $n$ factorial


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


The $1$st of the lucky numbers of Euler:
$n^2 + n + 2$ is prime for $n = 0$


The $1$st term of Göbel's sequence after the $0$th term $1$:
$2 = \paren {1 + 1^2} / 1$


The $1$st term of the $3$-Göbel sequence after the $0$th term $1$:
$2 = \paren {1 + 1^3} / 1$


The $1$st Sierpiński number of the first kind:
$2 = 1^1 + 1$


The $1$st prime Sierpiński number of the first kind:
$2 = 1^1 + 1$


The $1$st even number which cannot be expressed as the sum of $2$ composite odd numbers


The $1$st positive integer which divides the sum of all smaller primes:
$0 = 0 \times 2$
(there are no primes smaller than $2$)


The $1$st even integer that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes


The $1$st positive integer which is not the sum of $1$ or more distinct squares


The $1$st positive integer which cannot be expressed as the sum of distinct pentagonal numbers


The $1$st prime number of the form $n^2 + 1$:
$2 = 1^2 + 1$


The $1$st of $4$ numbers whose letters, when spelt in French, are in alphabetical order:
deux


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


The $1$st primorial which can be expressed as the product of consecutive integers:
$2 \# = 2 = 1 \times 2$


The $1$st of the $2$nd pair of consecutive integers whose product is a primorial:
$2 \times 3 = 6 = 3 \#$


The $1$st tri-automorphic number:
$2^2 \times 3 = 1 \mathbf 2$


The $1$st Euclid prime:
$2 = p_0\# + 1 = 1 + 1$


The $1$st (trivially) two-sided prime


The $1$st prime number consisting (trivially) of a string of consecutive ascending digits


The $1$st Hardy-Ramanujan number: the smallest positive integer which can be expressed as the sum of $2$ cubes in (trivially) $1$ way:
$2 = \map {\operatorname {Ta} } 1 = 1^3 + 1^3$


$2$nd Term

The $2$nd Catalan number after $(1)$, $1$:
$\dfrac 1 {2 + 1} \dbinom {2 \times 2} 2 = \dfrac 1 3 \times 6 = 2$


The $2$nd Ulam number after $1$


The $2$nd positive integer after $1$ such that all smaller positive integers coprime to it are prime


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


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


The $2$nd highly composite number after $1$:
$\map \tau 2 = 2$


The $2$nd special highly composite number after $1$


The $2$nd highly abundant number after $1$:
$\map \sigma 2 = 3$


The $2$nd superabundant number after $1$:
$\dfrac {\map \sigma 2} 2 = \dfrac 3 2 = 1 \cdotp 5$


The $2$nd almost perfect number after $1$:
$\map \sigma 2 = 3 = 4 - 1$


The $2$nd factorial after $1$:
$2 = 2! = 2 \times 1$


The $2$nd superfactorial after $1$:
$2 = 2\$ = 2! \times 1!$


The $2$nd factorion base $10$ after $1$:
$2 = 2!$


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


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


The $2$nd of the $5$ known powers of $2$ whose digits are also all powers of $2$:
$1$, $2$, $\ldots$


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


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


The $2$nd Bell number after $(1)$, $1$


The $2$nd positive integer after $1$ whose cube is palindromic (in this case trivially):
$2^3 = 8$


The $2$nd of the $1$st pair of consecutive integers whose product is a primorial:
$1 \times 2 = 2 = 2 \#$


$3$rd Term

The $3$rd integer $m$ after $0$, $1$ such that $m! + 1$ (its factorial plus $1$) is prime:
$2! + 1 = 2 + 1 = 3$


The $3$rd integer after $0$, $1$ such that its double factorial plus $1$ is prime:
$2!! + 1 = 3$


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


The $3$rd subfactorial after $0$, $1$:
$2 = 3! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} }$


The $3$rd integer $n$ after $-1$, $0$ such that $\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$ for integer $m$:
$\dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3 = 2^2$


The $3$rd integer $m$ after $0$, $1$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
$2^2 = \dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3$


The $3$rd palindromic integer after $0$, $1$ which is the index of a palindromic triangular number
$T_2 = 3$


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


The $3$rd integer $n$ after $0$, $1$ such that $5^n$ contains no zero in its decimal representation:
$5^2 = 25$


The $3$rd integer $n$ after $0$, $1$ such that both $2^n$ and $5^n$ have no zeroes:
$2^2 = 4, 5^2 = 25$


The $3$rd integer after $0$, $1$ which is palindromic in both decimal and ternary:
$2_{10} = 2_3$


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


Miscellaneous

The number of pairs of twin primes less than $10$:
$\tuple {3, 5}$, $\tuple {5, 7}$


Arithmetic Functions on $2$

\(\displaystyle \map \tau { 2 }\) \(=\) \(\displaystyle 2\) $\tau$ of $2$
\(\displaystyle \map \phi { 2 }\) \(=\) \(\displaystyle 1\) $\phi$ of $2$
\(\displaystyle \map \sigma { 2 }\) \(=\) \(\displaystyle 3\) $\sigma$ of $2$


Also see

Previous in sequence: $0$


Previous in sequence: $1$

Next in sequence: $3$


Next in sequence: $4$


Next in sequence: $5$


Next in sequence: $6$


Next in sequence: $7$


Next in sequence: $8$ and above


Next in sequence: $9$ and above


Next in sequence: $1$


Next in sequence: $3$


Next in sequence: $4$


Next in sequence: $5$


Next in sequence: $6$


Next in sequence: $7$


Next in sequence: $8$


Next in sequence: $9$ and above


Historical Note

The number $2$ (two) is understood to have been treated as a special case of a number from the earliest historical records.

Many early languages have specific forms of nouns for when two of an object are under consideration, as well as different forms for singular and plural.


To the ancient Greeks, in addition to having problems with the idea that $1$ is a number, it was questionable whether or not two ($2$) was actually a number either:

While it has a beginning and an end, it has no middle
Multiplication by $2$ consists merely of adding a number to itself, and multiplication was expected to do more than just add.

Thus $2$ was an exceptional case.


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 $2$ was considered to be the first male number, and said to personify the principle of diversity.

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


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


The dyad, as such, is never specifically defined in Euclid's The Elements, but introduced without definition in Power of Two is Even-Times Even Only.

In the words of Euclid:

Each of the numbers which are continually doubled beginning from a dyad is even-times even only.

(The Elements: Book $\text{IX}$: Proposition $32$)


Linguistic Note

There are many words in natural language which indicate a twofold collection:

dual, duel, brace, couple, pair, double and twin

for example.


Sources