# 48

This number is a composite.

If *n* is greater than or equal to 48, then there exists a prime between *n* and 9*n*/8, exclusive.

The smallest even number that can be expressed as a sum of two primes in five different ways (5 + 43, 7 + 41, 11 + 37, 17 + 31, 19 + 29). [Rivera]

48 = π(4)! * π(8)! [Firoozbakht]

48!_{2} + prime(84) is prime. [Farzannia]

48 is the smallest number such that (7^48+48) and (7^48-48) are both prime. [Bajpai]

Smallest number n such that 7^n+n and 7^n-n, both are prime. [Bajpai]

The smallest possible sum for a set of four distinct primes such that the sum of any three is prime: {5, 7, 17, 19}.

Questions: Is there a prime quadruplet (of the form {p, p+2, p+6, p+8}) with this property? (Click here for answer.) How about prime sextuplets, where the sum of any five are prime? (Click here for answer.) Or greater admissible prime constellations (*k*-tuples) such that the sum of any *k*-1 primes is prime? Update: Jens Kruse Andersen has found that due to divisibility by small primes, there is no *k* from 7 to 50 for which there exists a prime *k*-tuplet such that the sum of any *k*-1 is prime.

7^48+48 is the largest non-titanic prime of form 7^n+n. [Loungrides]

If n is greater than or equal to 48, then there exists at least one prime between n and 9n/8, exclusive. Proof by Robert Hermann Breusch in 1932. [Schott]

According to Li, Fang, and Kuo, there are 48 integers that serve as genes of primes.