0

Number
$0$ (zero) is:


 * The only number which is both positive and negative


 * The $1$st subfactorial:
 * $0 = 1! \paren {1 - \dfrac 1 {1!} }$


 * The $1$st integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
 * $0! + 1 = 1 + 1 = 2$


 * The $1$st integer such that its double factorial plus $1$ is prime:
 * $0!! + 1 = 2$


 * The $0$th Fibonacci number


 * The $1$st integer $n$ such that $2^n$ contains no zero in its decimal representation:
 * $2^0 = 1$


 * The $1$st integer $n$ such that both $2^n$ and $5^n$ have no zeroes:
 * $2^0 = 1, 5^0 = 1$


 * The $1$st integer which is palindromic in both decimal and binary:
 * $0_{10} = 0_2$


 * The $1$st integer which is palindromic in both decimal and ternary:
 * $0_{10} = 0_3$


 * The $1$st number which is (trivially) the sum of the increasing powers of its digits taken in order:
 * $0^1 = 0$


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


 * The $1$st integer $m$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
 * $0^2 = \dbinom {-1} 0 + \dbinom {-1} 1 + \dbinom {-1} 2 + \dbinom {-1} 3$


 * The $0$th triangular number:
 * $0 = \dfrac {0 \paren {0 + 1} } 2$


 * The $1$st of the $5$ Fibonacci numbers which are also triangular


 * The $1$st palindromic triangular number


 * The $1$st palindromic integer which is the index of a palindromic triangular number
 * $T_0 = 0$


 * The $1$st Ramanujan-Nagell number:
 * $0 = 2^0 - 1 = \dfrac {0 \paren {0 + 1} } 2$


 * The $1$st of the $3$ Fibonacci numbers which equals its index


 * The $1$st integer equal to the sum of the digits of its cube:
 * $0^3 = 0$

Linguistic Note
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