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We all know of Unsolved problems, like Goldbach,Legendre, and Grimm's conjectures.

Goldbach has the necessary condition of: There exists a prime between $n$ and $2n-2$, which means prime gaps are limited to $n-3$

Legendre has a necessary condition of: Ratio's of primes never exceed $\left(\frac{n+2}{n} \right)^2$

Grimm's has a necessary condition ( infinitely often) of: A prime gap never exceeds in length the number of primes less than the prime proceeding the gap.

My Question is: What other unsolved problems, have necessary conditions on the size of prime gaps?

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    $\begingroup$ There's the "obvious" one of the twin prime conjecture requiring an infinite number of prime gaps of $2$. $\endgroup$
    – John Omielan
    Commented Mar 1, 2019 at 17:28

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I like problems, like the twin prime conjecture, that are simple to state and understand.

One of my favourites that I think relates to what your question is asking is that of Professor Dr Dorin Andrica of Babeș-Bolyai University in Romania. He conjectures that, for all natural n; $$\sqrt{p_{n+1}}-\sqrt{p_n} \lt1$$

This has been unsolved since 1985, when it was proposed. It seems to be a very hard open problem in Number Theory.

Around 2004, Dan Grecu, had verified this for $p_n \lt 1000000$.

Looking it up just now, Imran Ghory has used data on the largest prime gaps to confirm the conjecture for $n$ up to $1.3002 × 10^{16}$.

I've googled this and there is a page on wikipedia here : https://en.wikipedia.org/wiki/Andrica%27s_conjecture

The Prof has a website of his many papers and books here; http://www.dorinandrica.ro/index.php

In American Mathemathics Monthly (1976) 61 it is given as a difficult unsolved problem that,

$$\lim_{n\to\infty} \big(\sqrt{p_{n+1}}-\sqrt{p_n}\big)=0$$

I've extracted this final piece of information from Richard Guys excellent book Unsolved Problems in Number Theory (third edition) which is still available : https://www.amazon.co.uk/Unsolved-Problems-Number-Intuitive-Mathematics/dp/0387208607

Incidentally, an excellent 'lighter' read on the fast moving developments over the last five years on the twin prime conjecture are beautifully described in Vicky Neale's book "Closing The Gap" : https://www.amazon.co.uk/Closing-Gap-Quest-Understand-Numbers/dp/0198788282

Much written before 2013 about the twin prime conjecture needs heavily rewriting is in the light of recent developments.

PS : There's a failed attempt at proving Andrica's conjecture here : An Approach to solve Andrica's Conjecture

And a more serious attempt here: Proof of Andrica when Assuming Oppermann

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    $\begingroup$ This I should have known about... it's Almost like extending Legendre tothe real values of n. $\endgroup$
    – user645636
    Commented Mar 13, 2019 at 19:59
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    $\begingroup$ @Roddy MacPhee : I'm glad it hit the mark. I enjoyed looking up the recent materials on it, so it was a good question in that it prompted me to I learn a few things from answering it. $\endgroup$
    – Martin Hansen
    Commented Mar 13, 2019 at 20:05
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    $\begingroup$ TPC can be rewritten as a statement of Goldbach partitions, or a statement on the naturals. $\endgroup$
    – user645636
    Commented Mar 13, 2019 at 22:23

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