# 31

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## Number

$31$ (thirty-one) is:

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

The $2$nd of $29$ primes of the form $2 x^2 + 29$:
$2 \times 1^2 + 29 = 31$ (Previous  ... Next)

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 $2$nd 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 $8$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 Marin Mersenne 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:
$2^{31} = 2 \, 147 \, 483 \, 648$

Expressible as the sum of successive powers starting from $1$ in in $2$ different ways:
$31 = 1 + 5 + 5^2 = 1 + 2 + 2^2 + 2^3 + 2^4$