59

Number

$59$ (fifty-nine) is:

The $17$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$

One of the prime factors of one of the terms of one of the $2$ representations of $635 \, 318 \, 657$ as the sum of $2$ fourth powers:

$635 \, 318 \, 657 = 59^4 + 158^4$

The upper end of the $4$th record-breaking gap between twin primes:

$59 - 43 = 16$

The $6$th safe prime after $5$, $7$, $11$, $23$, $47$:

$59 = 2 \times 29 + 1$

The smaller of the $7$th pair of twin primes, with $61$

The $7$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$

The $9$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$, $53$ such that the Mersenne number $2^p - 1$ is composite

The $10$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$

The $11$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$ such that $\ds m = \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime

The $27$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime

$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $\ldots$

The $34$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $44$, $45$, $46$, $49$, $50$, $54$, $55$ which cannot be expressed as the sum of distinct pentagonal numbers.