Monday 2 January

Today is 2 day of the year.
2 is the only even prime number.
For a polyhedron the number of faces plus the number of vertices minus the number of edges equals 2.
There is no prime between n! + 2 and n! + n (I am not aware of a proof for this to which I can link).
e.g.
10! + 2 = 3,628,802, 10! + 10 = 3,628,810
3,628,802 = 2 x 23 x 78,887
3,628,803 = 3 x 17 x 71,153
3,628,804 = 2^2 x 53 x 17,117
3,628,805 = 5 x 293 x 2,477
3,628,806 = 2 x 3 x 604,801
3,628,807 = 7 x 13 x 39,877
3,628,808 = 2^3 x 453,601
3,628,809 = 3^2 x 191 x 2,111
3,628,810 = 2 x 5 x 19 x 71 x 269
A corollary of this is that there are gaps between primes of any size. If one wanted to find a gap of at least m, then there is one between (m + 2)! + 2 and (m + 2) ! + (m + 2) since this is the previous proposition with n = m + 2 and
((m + 2) ! + (m + 2)) - ((m + 2)! + 2)
= (m + 2)! - (m + 2)! + (m + 2) - 2
= m as required.