# 53

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## Number

$53$ (fifty-three) is:

The $16$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$

The $1$st prime number which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$.

The $1$st prime number $p$ the period of whose reciprocal is $\dfrac {p - 1} 4$:
$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$

The $2$nd balanced prime after $5$:
$53 = \dfrac {47 + 59} 2$

The $5$th term of the $1$st $5$-tuple of consecutive integers have the property that they are not values of the $\sigma$ function $\map \sigma n$ for any $n$:
$\tuple {49, 50, 51, 52, 53}$

The $7$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$:
$53$, $5$, $3$ are prime

The $8$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$:
$2 \times 53 + 1 = 107$, which is prime.

The $8$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$ such that the Mersenne number $2^p - 1$ is composite

The $17$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$, $36$, $38$, $47$, $48$:
$53 = 6 + 47$

The $20$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $\ldots$

The $25$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $\ldots$

The probability that out of $53$ people, no $2$ of them share the same birthday is approximately $\dfrac 1 {53}$.