# 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
- $1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $\ldots$

## Also see

- Smallest Prime Number not Difference between Power of 2 and Power of 3
- Cyclic Permutations of 5-Digit Multiples of 41

*Previous*: Euler Lucky Number*Previous ... Next*: Proth Prime*Previous ... Next*: Prime Number Race between 4n+1 and 4n-1

*Previous ... Next*: Sum of Sequence of Alternating Positive and Negative Factorials being Prime*Previous ... Next*: Minimal Prime

*Previous ... Next*: Sequence of Fibonacci Numbers ending in Index*Previous ... Next*: Sophie Germain Prime

*Previous ... Next*: Twin Primes

*Previous ... Next*: Sequence of Indices of Composite Mersenne Numbers*Previous ... Next*: Prime Number*Previous ... Next*: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares*Previous ... Next*: Sequence of Integers whose Factorial plus 1 is Prime

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $41$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $41$