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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) 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 singledigit 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, n2], 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 (n  2)  (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 101^297*103 = 7# = 210. Does there exist another solution p,q,r q^2p*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,n1)? [Honaker]
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