7

Definition
$7$ (Seven) is:


 * The fourth prime number after $2, 3, 5$


 * The smallest prime number of the form $6 n + 1$:
 * $7 = 6 \times 1 + 1$


 * The start of an arithmetic progression of $6$ prime numbers:
 * $7, 37, 67, 97, 127, 157$


 * The first of the smallest pair of consecutive prime numbers different by $4$


 * The second Mersenne number and Mersenne prime after $3$, leading to the second perfect number $28$:
 * $7 = 2^3 - 1$


 * The smallest number that is not the sum of at most $3$ square numbers: see Integer as Sum of Three Squares


 * The third prime number after $2$ and $3$ to be of the form $n! + 1$ for integer $n$:
 * $3! + 1 = 6 + 1 = 7$
 * where $n!$ denotes $n$ factorial


 * The third $n$ after $4$ and $5$, and the largest known, such that $n! + 1$ is square: see Brocard's Problem:
 * $7! + 1 = 5040 + 1 = 5041 = 71^2$


 * The $1$st prime number whose period is of maximum length:
 * $\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$


 * The $2$nd of $5$ primes of the form $2 x^2 + 5$:
 * $2 \times 1^2 + 5 = 7$

Also see

 * Brocard's Problem