# 120

This number is a composite.

The largest integer *n* for which π(*n*) = *n*/4. [Woods]

120 = 3^{1} + 3^{2} + 3^{3} + 3^{4}. [Sladcik]

120^120-119^119 is prime. [Dobb]

120 is the smallest composite number of the form x! whose digit sum and x are primes. [Capelle]

All primes (except 2 and 3) are of form 6*n +/- 1. Note that 120 = 6*20 is the first number such that 6n+1 or 6n-1 does not contain a prime. [Luhn]

120 is the smallest three-digit number that may be expressed as the difference between the squares of consecutive primes in two different ways (120 = 31^2 - 29^2 = 17^2 - 13^2). [King]

120 is the first integer that can be represented as the sum of two odd primes in twelve different ways illustrating Goldbach's Conjecture. (120 = 7 + 113; 120 = 11 + 109; 120 = 13 + 107; 120 = 17 + 103; 120 = 19 + 101; 120 = 23 + 97; 120 = 31 + 89; 120 = 37 + 83; 120 = 41 + 79; 120 = 47 + 73; 120 = 53 + 67; 120 = 59 + 61) [Schiffman]

The smallest possible sum for a set of six distinct double-digit primes such that the sum of any five is prime: {11, 13, 17, 19, 23, 37}. [Loungrides]

The smallest number whose the sum of the reciprocals of its divisors is an odd prime. [Loungrides]

The decimal value of the fraction 135803690233087629631/135665296433469, to 120 digits, is 1001020.11202022102022102111203011122011203012102121102121103012112011302012112021103111103111112021202013102021202111113. Note that the 2nd, 3rd, 5th, 7th, 11th... digits are zeros. In fact, this 120-digit string has a 0 precisely in those positions corresponding to a prime number. This fraction has just 36 digits, yet its decimal value correctly encodes the primeness of the first 120 integers. Also note that 122223333432100/122098779 + 1/(10^7 - 1) gives the same 120-digit decimal, and uses only 29 digits. [Keith]