# 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 $2$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 $5$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.

## Also see

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

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## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $29$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $29$