# 23

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## Number

$23$ (**twenty-three**) is:

- The $1$st of the only two positive integers needing as many as $9$ positive cubes to express it:
- $23 = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3$

- The other is $239$.

- The $1$st element of the $1$st pair of consecutive prime numbers which differ by $6$:
- $23$, $29$: $29 - 23 = 6$

- The $1$st integer greater than $1$ which is not the sum of two Ulam numbers.

- The $2$nd prime $p$ after $11$ such that the Mersenne number $2^p - 1$ is composite

- The $2$nd prime number after $5$ of the form $n! - 1$ for integer $n$:
- $23 = 4! - 1$

- where $n!$ denotes $n$ factorial

- The $3$rd Woodall number after $1$, $7$, and $2$nd Woodall prime after $7$:
- $23 = 3 \times 2^3 - 1$

- The $3$rd Thabit number after $(2)$, $5$, $11$, and $4$th Thabit prime:
- $23 = 3 \times 2^3 - 1$

- The index of the $3$rd repunit prime after $R_2$, $R_{19}$:
- $R_{23} = 11 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$

- The $3$rd integer after $1$, $22$ which equals the number of digits in its factorial:
- $23! = 25 \, 852 \, 016 \, 738 \, 884 \, 976 \, 640 \, 000$

- which has $23$ digits

- The $4$th positive integer $n$ after $5$, $11$, $17$ such that no factorial of an integer can end with $n$ zeroes.

- The $4$th safe prime after $5$, $7$, $11$:
- $23 = 2 \times 11 + 1$

- The $4$th long period prime after $7$, $17$, $19$:
- $\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$

- The $5$th Sophie Germain prime after $2$, $3$, $5$, $11$:
- $2 \times 23 + 1 = 47$, which is prime.

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

- The $5$th prime number after the trivial $2$, $3$, $5$, $7$ consisting of a string of consecutive ascending digits

- The $5$th of $6$ integers after $2$, $5$, $8$, $12$ which cannot be expressed as the sum of distinct triangular numbers

- The $6$th happy number after $1$, $7$, $10$, $13$, $19$:
- $23 \to 2^2 + 3^2 = 4 + 9 = 13 \to 1^2 + 3^2 = 1 + 9 = 10 \to 1^2 + 0^2 = 1$

- The $9$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$

- The $9$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
- $3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $\ldots$

- The $12$th positive integer which is not the sum of $1$ or more distinct squares:
- $2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $\ldots$

- The $12$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
- $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $\ldots$

## Also see

- Numbers not Expressible as Sum of Less than 9 Positive Cubes
- Smallest Integer not Sum of Two Ulam Numbers
- Square-Bracing Problem: Non-Crossing Rods

*Previous ... Next*: Two-Sided Prime*Previous ... Next*: Woodall Number*Previous ... Next*: Prime Numbers Composed of Strings of Consecutive Ascending Digits*Previous ... Next*: Woodall Prime

*Previous ... Next*: Sequence of Indices of Composite Mersenne Numbers*Previous ... Next*: Sophie Germain Prime*Previous ... Next*: Safe Prime*Previous ... Next*: Thabit Number*Previous ... Next*: Thabit Prime

*Previous ... Next*: Happy Number*Previous ... Next*: Prime Number*Previous ... Next*: Long Period Prime*Previous ... Next*: Index of Repunit Prime

*Previous ... Next*: Numbers not Sum of Distinct Squares*Previous ... Next*: Numbers Equal to Number of Digits in Factorial*Previous ... Next*: 91 is Pseudoprime to 35 Bases less than 91

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $23$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $23$