3

Number
$3$ (three) is:


 * The $2$nd prime number after $2$


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


 * The $1$st odd prime


 * The $1$st of the $1$st pair of twin primes, with $5$


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


 * The $2$nd triangular number after $1$:
 * $3 = 1 + 2 = \dfrac {2 \left({2 + 1}\right)} 2$


 * 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:
 * $3 = 2^{\left({2^0}\right)} + 1$


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


 * 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 $1$st prime number which is the first of a pair of twin primes:
 * $3, 5$


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


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


 * The $3$rd highly abundant number after $1, 2$:
 * $\sigma \left({3}\right) = 4$


 * The $2$nd Stern number after $1$.


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


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


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


 * The $1$st of the sequence of $n$ such that $p_n \# - 1$, where $p_n \#$ denotes primorial of $n$, is prime:
 * $p_3 \# - 1 = 2 \times 3 - 1 = 5$


 * The $2$nd of the sequence of $n$ such that $p_n \# + 1$, where $p_n \#$ denotes primorial of $n$, is prime, after $2$:
 * $p_3 \# + 1 = 2 \times 3 + 1 = 7$


 * The $1$st of an arithmetic progression of primes:
 * $3, 5, 7$


 * The sum of the $1$st two factorials:
 * $3 = 1! + 2!$


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


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


 * The $2$nd number after $1$ whose $\sigma$ value is square:


 * 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 $3$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 $2$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 $\tau$ function equals its $\sigma$ function:
 * $\phi \left({3}\right) \tau \left({3}\right) = 2 \times 2 = 4 = \sigma \left({3}\right)$


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


 * The $2$nd palindromic triangular number after $1$.


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


 * The $2$nd positive integer solution after $1$ to $\phi \left({n}\right) = \phi \left({n + 1}\right)$:
 * $\phi \left({3}\right) = 2 = \phi \left({4}\right)$


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


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


 * The $2$nd element of the Fermat set after $1$.


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


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


 * 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 $\tau \left({n}\right) \mathrel \backslash \phi \left({n}\right) \mathrel \backslash \sigma \left({n}\right)$:
 * $\tau \left({3}\right) = 2$, $\phi \left({3}\right) = 2$, $\sigma \left({3}\right) = 4$

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