Gödel proved that there are true sentences that cannot be proved.
Suppose I told you that the Goldbach conjecture is one of those. (The Goldbach conjecture supposes that every even integer number can be expressed as the sum of two odd primes.)
Is that logically possible? (And, no, I haven't proved it!)
Primise 1: Prime numbers seem to apear at random intervals.
Premise 2: If premise one is True, then three would exist at some point a range of consectutive non prime numbers numbers that included both x and 2x.
Prime 3: if premise 2 is true then there exist an even number 2x that cannot be oken into two prime numbers ( all prime numbers to this poinr are less then x). This would disprove the Goldbach conjecture.
Therefor:: either the Goldbach conjecture is false or prime numbers are not at random intervals.