# 210

This number is a composite.

The primes 47, 257, 467, 677, 887, 1097 and 1307 constitute a progression of 7 terms with a common difference of 210. [Barrow , Bush and Taylor]

210 is the smallest number with 4 distinct prime divisors.

The largest single-digit primorial value (7# = 2 * 3 * 5 * 7 = 210). [Nicholson]

It has been estimated that 210 becomes a "jumping champion" at around 10^425.

(21, 20, 29) and (35, 12, 37) are the two least primitive Pythagorean triangles with different hypotenuses and the same area (=210). [Sierpinski]

The number of distinct representations of a number n as the sum of two primes is at most the number of primes in the interval [n/2, n-2], and 210 is the largest value of n for which this upper bound is attained. In other words, 210 is the largest positive integer n that can be written as the sum of two primes in π(n - 2) - π(n/2 - 1) distinct ways. Reference: An upper bound in Goldbach's problem. [Capelle]

210^k+1 is prime for k = 2, 1, 0. [Bajpai]

Bergot's Problem: Let p,q,r be three consecutive primes and note that |101^2-97*103| = 7# = 210. Does there exist another solution p,q,r |q^2-p*r| that equals a larger primorial?

C(10,4) = 2*3*5*7 = 7# = 210, i.e., the number of combinations of 10 objects taken 4 at a time is a primorial. Does there exist a larger example that ignores C(n,1) and C(n,n-1)? [Honaker]

210 is the product of the first (2!+1!+0!) primes. [Worrom]