23

Number
$23$ (twenty-three) is:


 * The number of unit rods required to brace a unit square when rods may not cross.


 * 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 of $5$ primes of the form $2 x^2 + 5$:
 * $2 \times 3^2 + 5 = 23$


 * 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