24

This number is a composite.

The smallest number which is the sum of two distinct odd primes in three ways, i.e., 24 = 5 + 19 = 7 + 17 = 11 + 13. [McCranie]

All even numbers up to 24 are expressible as an algebraic sum of the first six odd primes. [Rivera]

24²⁵⁰ * 250²⁴ + 1 and 24¹²⁵ * 125²⁴ - 1 are primes. [Kulsha]

24 divides the difference between any two prime squares greater than three. [Walser]

There is no prime in which the mean gap between the first n successive primes is exactly 24. [Dodson]

In base 24, the square of any number relatively prime to the base (i.e., in this case, any number not divisible by 2 or 3) will end in the digit "1", and 24 is the largest base having this property. [Kerby]

24 divides p^2-1 for any prime greater than three. [Hockney]

The twenty-fourth Fermat number (2^2^24 + 1) is composite. [Crandall , Mayer and Papadopoulos]

24 is the largest number less than a thousand such that if it is raised to the sum of its digits plus the sum of its digits, plus and minus one, the results are twin primes. I.e., 24^(2+4)+(2+4) ± 1 are twin primes. [Opao]

The atomic numbers (proton-electron pairs) of 24 of the naturally occurring (Hydrogen through Uranium) elements from the periodic table are prime. Coincidentally, 24 of the naturally occurring elements have a prime number of nucleons (protons and neutrons). [Welch]

(24!)^2 + 1 is prime. [Patterson]

The 24th prime number is 89. The sum of the first 24 prime numbers is 963. Note that 89 mod 24, 963 mod 24, 963 mod 89, 963 mod (89 + 24), 963 mod (89 - 24), 963 mod (89 mod 24), and the sum of the prime factors of 963 are all prime! [Ranta]

π(24) = π(2!)*π(4!). [Firoozbakht]

prime(1)^prime(1) + prime(2)^prime(2) + ... + prime(24)^prime(24) is prime. The next such prime (if it exists) has more than 46000 digits. [Firoozbakht]

The largest known natural number n such that: prime(n+1) - prime(n) = φ(n). Note that prime(φ(24)+1) - prime(φ(24)) = φ(φ(24)). [Capelle]

If the fraction 4/n is not equal to a sum of three distinct unit fractions, then each prime divisor of n must be congruent to 1 modulo 24. [Post]

24*24+2424 plus/minus one are twin primes. [Silva]

Exactly half, 24 of the contiguous 48 states of the United States plus the District of Columbia (DC) are adjacent to a prime number of other states or districts. DC, FL, RI, SC, WA are connected to 2 other states; CA, CT, DE, LA, MI, ND, NH, NJ, VT are connected to 3 other states; GA, IL, MA, MD, NV, NY, OH, UT, WV are connected to 5 other states; KY is connected to 7 other states. [Post]

There are no known examples of constellations with 24 primes. [Anderson]

(24, 25) is the only case of two successive double-digit numbers n, n+1 such that the product of each of them with its reversal plus 1 is emirp, i.e., 24*42+1=1009, 25*52+1=1301. [Loungrides]

The largest number of primes that can be found on the Tetractys Puzzle.

The smallest non-sphenic number with exactly eight divisors.

24 is the smallest composite k such that the concatenation of the prime factors of k, with multiplicity, in some order is divisible by k. [Bergot and Israel]

OEIS A390417 has at most 24 terms, but whether or when the remaining rankings occur is not known.