# 23

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## 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 $2 x^2 + 5$:
$2 \times 3^2 + 5 = 23$ (Previous  ... Next)

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
$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $\ldots$