# When is the earliest large prime gap also the latest large prime gap?

Suppose one finds the earliest prime gap of at least a certain size $g$, so that $p_{n+1}-p_n=g$ and $n$ is the smallest index for which the gap is as big as $g$.

Now consider the relative size of the gap: $\dfrac{p_{n+1}-p_n}{p_n}$

When will $n$ be the largest index for which the relative size is that big?

Is the set of all values of $n$ for which that happens infinite?

Is the set of all values of $n$ for which that does not happen infinite?

(For example, this happens when $p_n=3$ and when $p_n = 113$.)

36 views and two votes, but complete silence ensued when I posted this on math.stackexchange.com. This time it's verbatim the same as what I put there: https://math.stackexchange.com/questions/870438/when-is-the-earliest-large-prime-gap-also-the-latest-large-prime-gap

PS in response to comments below: Maybe I should be more explicit. The number $113$ is the smallest prime for which the absolute gap size, $127-113=14$, is $\ge14$, and if I'm not mistaken, is also the largest prime for which the relative gap size $(127-113)/113=0.12389\ldots$, is $\ge(127-113)/113=0.12389\ldots$.

So consider three sets:

• values of $n$ are that are simultaneously the smallest for which the absolute gap size is $\ge$ some number and the largest for which the relative gap size is $\le$ some number;

• values of $n$ that are the smallest for which the absolute gap size is $\ge$ some number but not the largest for which the relative gap size is $\le$ whichever value they take on;

• values of $n$ that are the largest for which the relative gap size is $\le$ some number but not the smallest for which the absolute gap size is $\ge$ whichever number they take on.

So the question would be which of these are empty, which are non-empty but finite, and which are infinite.

• Even though the current knowledge of good bounds on maximal gaps is small, I think enough is known since 1931 (Westzynthius showing that the relative size is greater than average for infinitely many n by a ratio of something like loglogn) that your two sets n are both infinite. (It may even follow from older results.) – The Masked Avenger Jul 20 '14 at 3:53
• @TheMaskedAvenger : I'm not seeing that your comment actually answers the questions. If I'm not mistaken, in the case of $113$, where the relative gap size, $(127-113)/113=14/113\approx 0.12389\ldots$ is bigger than any that occurs later. I have to suspect that "bigger than average" means bigger than some "average" that decreases as $n$ grows. – Michael Hardy Jul 20 '14 at 4:12
• I notice that mathoverflow has no "prime gaps" tag. Stackexchange has that. Should that be created? – Michael Hardy Jul 20 '14 at 4:27
• We don't have the knowledge of when the next maximal gap occurs ( after the ones that have been tabulated for primes below 10^17 or therabouts). We do know that the growth is sublinear, from Hoheisel and others, so lim sup of the relative size is decreasing. I think this means the first of your three sets is infinite. I suspect work of Westzynthius et al shows that the union of your other two sets is infinite, but I am less sure of that. – The Masked Avenger Jul 20 '14 at 5:30
• What do you mean by saying that the lim sup of the relative size is decreasing? The relative size is $(p_{n+1}-p_n)/p_n$. That is a sequence, and it has a lim sup, and its lim sup is a number, not a sequence. What would it mean to say a number is decreasing? (And I'll be very surprised if you tell me the lim sup is not $0$.) – Michael Hardy Jul 20 '14 at 22:43