4

Conjecture

If we have two consecutive prime numbers $p_{n}$ and $p_{n+1}$, and another arbitrary prime number $p_a$ such that $p_{n} < p_{n+1} < p^2_{a}$, then it follows that $p_{n+1} - p_{n} < p_{a} $.

Are there any known counter examples or reasons for this being true?

What is the name of this property or a similar one with respect to general sequences, (not necessarily the sequence of primes)?

Brad Graham
  • 1,171
  • 2
    $p_n=113$, $p_{n+1}=127$, $p_a=13$ is a counterexample and my gut feeling tells me there wouldn't be any others. The Wikipedia entry on prime gaps provides quite a few conjectures, some of which are stronger than this one. – Peter Košinár Feb 24 '14 at 16:36
  • Are you familiar with Opperman's Conjecture on prime gaps, and related results? See also this question. – Bill Dubuque Feb 24 '14 at 16:48
  • No not really, I tend to "explore" without following reference material, but I probably should. So thanks for giving me a target topic to get up to grasp with. Thanks for the counter example Peter, 13 seems to be my unlucky number as it was also a counter example to another belief I had: link. PS. that was my question Bill! – Brad Graham Feb 24 '14 at 18:09

1 Answers1

2

Probably the only counterexample is (113, 127, 13) as pointed out by Peter Košinár. Such a gap would be larger than the square root of $p_{n+1}$ and so it is easy to check that there are no examples with $p_n<4\cdot10^{18}$ using existing tables of maximal prime gaps. If there is another example it must be a gap at least 46 million times the length of the average gap of at its size.

Charles
  • 32,999