Prime Number

Prime Number

Prove we can always find an interval [p, q] wherein no prime exists, for any p, q ∈ N
Proof:
For any n ∈ N
Let H= (n+1)! = 1 x 2 x ... x(n+1)
Let G = {H+2, H+3,. . . , H+(n+1) }
=> G composites (trivial)

Choose [p, q] such that  :
1. p the closest prime < H+2,
2. q the closest prime > H+(n+1)
=> no prime ∈ [p, q]

---n----------------n!-(H+1)-p[(H+2)---------H+(n+1)]q------------------