# 41

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## Number

$41$ (forty-one) is:

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

The $1$st prime number which is not the difference between a power of $2$ and a power of $3$.

The $4$th integer after $2$, $5$, $17$ at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied.

The $5$th Proth prime after $3$, $5$, $13$, $17$:
$41 = 5 \times 2^3 + 1$

The $5$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
$3$, $5$, $11$, $13$, $41$

The $5$th prime $p$ after $11$, $23$, $29$, $37$ such that the Mersenne number $2^p - 1$ is composite

The smallest positive integer the decimal expansion of whose reciprocal has a period of $5$:
$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$

The smaller of the $6$th pair of twin primes, with $43$

The $6$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$ such that the Fibonacci number $F_n$ ends in $n$

The $6$th integer after $7$, $13$, $19$, $35$, $38$ the decimal representation of whose square can be split into two parts which are each themselves square:
$41^2 = 1681$; $16 = 4^2$, $81 = 9^2$

The $6$th and largest lucky numbers of Euler after $2$, $3$, $5$, $11$, $17$:
$n^2 + n + 41$ is prime for $0 \le n < 39$.

The $7$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$:
$2 \times 41 + 1 = 83$, which is prime.

The $7$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$

The $8$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$

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

The $20$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $\ldots$