29

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 $2 x^2 + 29$:
 * $2 \times 0^2 + 29 = 29$


 * 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 $2 x^2 + 11$:
 * $2 \times 3^2 + 11 = 29$


 * 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 $5$th long period prime after $7$, $17$, $19$, $23$:
 * $\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$


 * 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 $15$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
 * $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $\ldots$


 * 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.

Also see

 * Hilbert-Waring Theorem: Cubes
 * Smallest Integer not Sum of Two Ulam Numbers
 * Square-Bracing Problem: Non-Crossing Rods