# 105

This number is a composite.

Paul Erdős conjectured that this is the largest number *n* such that the positive values of *n* - 2^{k} are all prime.

105/π^{4} = (1 + 1/2^{4})(1 + 1/3^{4})(1 + 1/5^{4})(1 + 1/7^{4})(1 + 1/11^{4}) … . [Ramanujan]

The first product of 3 distinct odd primes p. Note that p +/- 2 are primes. [Gallardo]

105 is the first degree for which the cyclotomic polynom factor are not all 1, 0 or -1. [Cristian]

The largest natural number n for which all odd numbers k relatively prime to n, with 1 < k < n, are primes. It is also the largest for which 2n-p is prime for each prime p in [n, 2n-2]. [Capelle]

The only 3-digit number n whereby subtracting from it each of the first eight powers 2^n, (n=1 to 8), then taking the absolute values of the results, produce prime numbers, i.e., 105-2^1 = 103, 105-2^2 = 101, 105-2^3 = 97, 105-2^4 = 89, 105-2^5 = 73, 105-2^6 = 41, |105-2^7|= 23, |105-2^8| = 151. [Loungrides]