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Let $p$ denote an odd prime power (a prime power is a number of the form $p^k$ where $p$ is a prime and $k \ge 1$). These are the numbers OEIS A061345 without the $1$.

For $p \ge 5$ defined prev$(p)$ to be the (strict) predecessor of $p$ in the set of odd prime powers, and gap$(p)$$\, =\, p\, - $ prev$(p)$ (by convention gap$(3)$$\, =\, 0$).

For example, if $p\, =\, 1361$ then prev$(p)$$\, =\, 1331$ and gap$(p)$$\, =\, 30$.

Conjecture: Let $p,\, q$ be odd prime powers. If gap$(p)$$\, > \, $gap$(q)$ for all $q < p$ then $p$ is prime.

Does anyone have an idea how to prove (or disprove) this?

Numerical verification confirmed the conjecture for numbers $\le 42652618807$. In this range all the potential candidates fulfilled the statement and identified this way 42 prime numbers.

Another observation is that the sum of the reciprocals seems to converge towards a value that starts $0.3141\ldots$

Peter Luschny
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  • My first idea on how to disprove it is to check numerically up to some large value. – eyeballfrog Feb 02 '23 at 16:25
  • What have you tried? Clearly the first step is to do a good search. – lulu Feb 02 '23 at 16:26
  • @eyeballfrog I have checked up to 42652618807. – Peter Luschny Feb 02 '23 at 16:28
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    I suggest editing your post to include the results of your search. How many primes did you identify this way? – lulu Feb 02 '23 at 16:29
  • @lulu 42 primes. – Peter Luschny Feb 02 '23 at 16:30
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    And how many composite prime powers are on that list? Again, I suggest editing your post to include the results of the search...don't leave critical information for the comments. – lulu Feb 02 '23 at 16:34
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    I think it could be useful to give also this list of 42 primes, and the gap associated to each one. Not sure it will help, but it will interest people who read this. – Lourrran Feb 02 '23 at 16:39
  • @Lourrran Yes, I will do, as a sequence in the OEIS. – Peter Luschny Feb 02 '23 at 16:46
  • Concerning the constant in the last part, over which numbers are the reciprocals summed up ? – Peter Feb 03 '23 at 09:48
  • Let's define $g$ : $g(k) = min { x \in \mathbb{N}, gap(x)=k }$ and $h$ : $h(k) = min { x \in \mathbb{N}, gap(x) \ge k }$ ; did you check if these 2 functions are identical ? Can we have another conjecture : if $gap(p) \ne gap(q)$ for all $q<p$ then $p$ is prime ? – Lourrran Feb 03 '23 at 12:51
  • @Peter The sum runs over the reciprocals of the identified primes. – Peter Luschny Feb 04 '23 at 00:27
  • @Lourrran I do not know. But imagine a situation similar to [23, 25, 27, 29] where in the interval between two prime numbers there are two prime powers, but the distance between the prime powers is very large. Isn't it conceivable that this gap is larger than all the gaps observed so far? In this case, the larger prime power would be one of the numbers sought without being a prime number. – Peter Luschny Feb 04 '23 at 00:28
  • I think you have the answer to my first question. You have your 42 gaps. look at these 42 values. If last one is 45 ou something like this, my 2 functions are similar, if last one is 100 or higher, we can conjecture they are different. – Lourrran Feb 04 '23 at 08:18
  • I upvoted but it would be nice to see the jumping champions (or sometimes called high water marks) for the differences between the consecutive odd prime powers which also would give us an idea which difference we have to beat at all to find a counterexample. – Peter Feb 04 '23 at 14:28
  • I recalculated the jumping champions (but I won't post them, this is the job of the author ! ) and looked for the gaps occuring for non-prime prime powers below some threashhold I do not exactly remember (at least $10^{16}$) making a counterexample very unlikely. The best chance for a possible counterexample is a gap for the square of a prime. – Peter Feb 06 '23 at 16:39

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