31

Number
$31$ (thirty-one) is:


 * The $11$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$


 * The $1$st of the $2$ known numbers (with $8191$) expressible as the sum of successive powers starting from $1$ in in $2$ different ways:
 * $31 = 1 + 5 + 5^2 = 1 + 2 + 2^2 + 2^3 + 2^4$


 * The $2$nd of $29$ primes of the form $2 x^2 + 29$:
 * $2 \times 1^2 + 29 = 31$


 * The $2$nd prime number after $3$ that can be found starting from the beginning of the decimal expansion of $\pi$ (pi):
 * $3 \cdotp 1 (4159 \, 26535 \, 897 \ldots)$


 * The $3$rd emirp after $13$, $17$


 * The $3$rd Mersenne number and the $3$rd Mersenne prime after $3$, $7$:
 * $31 = 2^5 - 1$


 * The $3$rd Euclid number after $3$, $7$:
 * $31 = p_3\# + 1 = 2 \times 3 \times 5 + 1$


 * The $4$th Euclid prime after $2$, $3$, $7$:
 * $31 = p_3\# + 1 = 2 \times 3 \times 5 + 1$


 * The $4$th after $2$, $3$, $19$ of the $4$ known primes $p$ such that $\dfrac {p^p - 1} {p - 1}$ is itself prime:
 * $\dfrac {31^{31} - 1} {31 - 1} = 568 \, 972 \, 471 \, 024 \, 107 \, 865 \, 287 \, 021 \, 434 \, 301 \, 977 \, 158 \, 534 \, 824 \, 481$


 * The larger of the $5$th pair of twin primes, with $29$


 * The $6$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
 * $2$, $3$, $5$, $7$, $11$, $31$


 * The $7$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$


 * The $7$th minimal prime base $10$ after $11$, $13$, $17$, $19$, $23$, $29$


 * The $8$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$


 * The $8$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$:
 * $31 \to 3^2 + 1^2 = 9 + 1 = 10 \to 1^2 + 0^2 = 1$


 * The index of the $8$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$:
 * $M_{31} = 2^{31} - 1 = 2 \, 147 \, 483 \, 647$


 * The $9$th lucky number:
 * $1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $\ldots$


 * The index of the $9$th Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$, $17$, $19$ which asserted to be prime


 * The $16$th positive integer which is not the sum of $1$ or more distinct squares:
 * $2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $\ldots$


 * The $16$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
 * $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $\ldots$


 * The $20$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$, $19$, $20$, $21$, $24$, $25$, $26$, $29$, $30$ which cannot be expressed as the sum of distinct pentagonal numbers.


 * The $21$st integer $n$ such that $2^n$ contains no zero in its decimal representation:
 * $2^{31} = 2 \, 147 \, 483 \, 648$


 * The $23$rd (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $20$, $21$, $24$, $25$, $26$, $27$, $30$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

Also see

 * Square of Reversal of Small-Digit Number