20

This number is a composite.

The smallest composite number r such that n = 4r + 3 and m = 8r + 7 are primes. [Rivera]

Oliver Sacks' describes in his book 'The MAN Who MISTOOK HIS WIFE for a HAT,' a pair of autistic twin brothers who were swapping 20-figure primes.

The smallest number n such that neither 6n-1 nor 6n+1 are primes. [Necula]

20 ones plus 20 is prime. This is the largest such two-digit number. [Opao]

There are 20 books in the Bible that have a prime number of chapters. [Opao]

Reversal(1^20 + 2^20 + ... + 20^20) is prime. Reversal(20) is the only other known number with this property. [Firoozbakht]

20^20 - 11...1 (20 ones) and 20 + 11...1 (20 ones) are primes. [Firoozbakht]

The 20th palindromic prime contains a digit sum of 20. [Post]

The smallest composite number which, summed to all previous primes, yields a prime. [Silva]

π(20) = 2^3 + 0^3. It is the smallest number with this property. [Kumar]

20 is one of the numbers n such that sum of prime divisors of n is exactly one less than number of divisors of n and number of prime divisors of n exactly divides n. [Sivaraman]

The smallest number that cannot be either prefixed or followed by one digit to form a prime. [Homewood]

The 20th prime divides the sum of the first 20 primes. 20 is the smallest composite with this property. [Hasler]

The only number n < 1000, such that subtracting from it each of the first seven powers of 3, then all absolute differences that are obtained, 20-3^0, 20-3^1, …, 20-3^6 are primes, i.e., 19, 17, 11, 7, 61, 223, 709. [Loungrides]

e cubed is close to 20. [Meyes]

In the ninth book of Euclid's work “Elements,” in Proposition 20, there appears for the first time a mathematical proof of the theorem that there are infinitely many prime numbers. [Shalit]

The smallest number n such that 1n1, 3n3, 7n7, 9n9 are all prime. [Bajpai]

The smallest number n whose sum of digits equals omega(n).

Let n be a positive integer. Write the prime factorization in canonical (standard) form. For example, 120 = 2 ^ 3 * 3 * 5 where the primes are written in increasing order of magnitude, and exponents of 1 are omitted. We then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat. So, for example, f(120) = f(2 ^ 3 * 3 * 5) = 2335. Next, because 2335 = 5 * 467, it maps, under f, to 5467. Now f(5467) = f(7 *11 *71) = 71171. Since 71171 is prime, it maps to itself. Thus 120 → 2335 → 71171→...,so we have climbed to a prime, and we stop there forever. 20 is the first composite integer where this iterative factorization and concatenation procedure does not yet yield a prime after getting to more than one hundred digits. The initial ten iterations for 20 are 20 → 225 → 3252 → 223271 → 297699 → 399233 → 715623 → 3263907 → 32347303 → 160720129 → .... This was the fifth problem in John Horton Conway's list of prize problems. James Davis in 2017 proved that the Climb to a Prime conjecture was indeed false. The problem is a neat variation of The Home Prime Conjecture which remains open. [Schiffman]