# 29

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## Number

$29$ (twenty-nine) is:

The $10$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$

The $1$st of the $5$th pair of twin primes, with $31$

The $1$st of $29$ primes of the form $2 x^2 + 29$:
$2 \times 0^2 + 29 = 29$ (Next)

The $2$nd of the $1$st pair of consecutive prime numbers which differ by $6$:
$29 - 23 = 6$

The $2$nd primorial prime after $5$:
$29 = p_3 \# - 1 = 5 \# - 1 = 2 \times 3 \times 5 - 1$

The $2$nd after $21$ of the $5$ $2$-digit positive integers which can occur as a $5$-fold repetition at the end of a square number

The $3$rd number such that $2 n^2 - 1$ is square, after $1$, $5$:
$2 \times 29^2 - 1 = 2 \times 841 - 1 = 1681 = 41^2$

The $3$rd prime $p$ after $11$, $23$ such that the Mersenne number $2^p - 1$ is composite

The upper end of the $3$rd record-breaking gap between twin primes:
$29 - 19 = 10$

The $4$th of $11$ primes of the form $2 x^2 + 11$:
$2 \times 3^2 + 11 = 29$ (Previous  ... Next)

The $5$th Lucas prime after $2$, $3$, $7$, $11$.

The $5$th positive integer $n$ after $5$, $11$, $17$, $23$ such that no factorial of an integer can end with $n$ zeroes.

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

The $6$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$:
$2 \times 29 + 1 = 59$, which is prime.

The $7$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$, $18$:
$29 = 11 + 18$

The $12$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $\ldots$

The $18$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$, $16$, $19$, $20$, $21$, $24$, $25$, $26$ which cannot be expressed as the sum of distinct pentagonal numbers.