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.