# 21

The smallest number with 2 prime factors that do not divide 10.

2^{21} - 21 is prime. [Brown]

Blackjack primes are separated by exactly 21 consecutive composite numbers. Note that the pair {1129, 1151} is the smallest example.

The smallest prime of the form *n*^{21} + 21 begins with a decimal expansion of π = 3.141....

Phi(21) is the reversal of 21. [Honaker]

The 21st set of blackjack primes (primes with a gap of 21) is 11329 and 11351. [Fougeron]

The smallest triangular number whose sum of aliquot divisors is prime. [Gupta]

21 is the only number *m* such that *m* = π(π(π(*m*)!_{2}))). [Firoozbakht]

21 is the only known multidigit number m such that both numbers m * m! - prime(m) and m * m! + prime(m) are primes. [Firoozbakht]

The number of known positive integers which cannot be written as p * q + r, where p, q, r are three distinct primes. [Capelle]

The smallest semiprime that is a product of distinct Mersenne primes.

21 repeated twenty-one times, following 1, forms a smoothly undulating palindromic prime. [Silva]

21 followed by its reversal, plus/minus one, are twin primes. The first integer with that property. [Silva]

21 = π(2*21) + π(1*21). [Arabi]

The first Fibonacci semiprime is followed by two others (34 and 55). [Silva]

The only brilliant number n that creates two 4-digit emirps by inserting it between its prime factors, i.e., 3217, 7213. [Loungrides]

Is 21 the only Fibonacci number that falls between cousin primes (19 and 23)? [Gaydos]

Finding the fourth term in sequence A323604 will require counting the primes less than or equal to a 21-digit number.

The only known Fibonacci number that is the sum of three distinct Fibonacci primes, i.e., 21 = 3 + 5 + 13. [Honaker]