# 84

This number is a composite.

Its digits are composite. Yet each digit minus 1, the sum of the digit minus 1 and the absolute value of the difference of the digits minus 1 is prime. [Russo]

The product of its digits minus 1 is prime. [Wolfe]

84 is the smallest number that can be expressed as the sum of 3 distinct primes raised to distinct prime exponents
(2^{5} + 3^{3} + 5^{2}). [Russ]

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

84 = prime(prime(8))+prime(prime(4)). It is the only such number. [Firoozbakht]

The smallest integer that can be expressed as the sum of two primes in eight different ways: {5 + 79}, {11 + 73}, {13 + 71}, {17 + 67}, {23 + 61}, {31 + 53}, {37 + 47}, {41 + 43}. [Schiffman]

The smallest number that can be written in four ways as the sum of a set of four distinct primes such that the sum of any three terms in each set is prime: {5, 13, 23, 43}, {11, 13, 17, 43}, {11, 13, 23, 37}, {13, 17, 23, 31}. [Loungrides]