2

== 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 of the $4$ known primes $p$ such that $\dfrac {p^p - 1} {p - 1}$ is itself prime:
::$\dfrac {2^2 - 1} {2 - 1} = 3$
: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:
::$2 = p_1 \# = 2 \# := \ds \prod_{k \mathop = 1}^1 p_k = 2$
: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 prime number 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) left-truncatable prime
:The $1$st (trivially) right-truncatable prime
: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:
::$2 = \map {\operatorname {Ta} } 1 = 1^3 + 1^3$
:The $1$st of $6$ integers which cannot be expressed as the sum of distinct triangular numbers
:The $1$st minimal prime base $10$
:The number of pairs of twin primes less than $10^1$
=== $2$nd Term ===
:The $2$nd Catalan number after $(1)$, $1$:
::$2 = \dfrac 1 {2 + 1} \dbinom {2 \times 2} 2 = \dfrac 1 3 \times 6$
: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 $1$:
::$\map {\sigma_0} 2 = 2$
:The $2$nd special highly composite number after $1$
:The $2$nd highly abundant number after $1$:
::$\map {\sigma_1} 2 = 3$
:The $2$nd superabundant number after $1$:
::$\dfrac {\map {\sigma_1} 2} 2 = \dfrac 3 2 = 1 \cdotp 5$
:The $2$nd almost perfect number after $1$:
::$\map {\sigma_1} 2 = 3 = 4 - 1$
: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:
::$1 \times 2 = 2 = 2 \#$
:The index of the $2$nd Mersenne number after $1$ which asserted to be prime
:The $2$nd (strictly) positive integer after $1$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The $2$nd after $1$ of $6$ integers $n$ such that the alternating group $A_n$ is ambivalent
:The $2$nd after $1$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
:The $2$nd after $0$ of the $6$ integers which are the middle term of a sequence of $5$ consecutive integers whose cubes add up to a square
::$0^3 + 1^3 + 2^3 + 3^3 + 4^3 = 100 = 10^2$
:The number of integer partitions for $2$:
::$\map p 2 = 2$
=== $3$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 palindromic integer after $0$, $1$ whose square is also palindromic integer
::$2^2 = 4$
: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 $1$, $1$:
::$2 = 1 + 1$
:The number of distinct free trominoes
=== Miscellaneous ===
:The number of pairs of twin primes less than $10$:
::$\tuple {3, 5}$, $\tuple {5, 7}$
== Also see ==
=== Previous in sequence: $0$ ===
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== Historical Note ==
== Linguistic Note ==
== Sources ==
* : $2$
* : $2$
Category:Specific Numbers
Category:2