Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question is thus: which even positive integers are known to be both prime gaps (at least once) and the sum of 2 primes? Can we prove that the set of such integers has asymptotic density $1$ among all even positive integers?
Thanks in advance.
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asked Dec 20, 2015 at 14:17
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$\begingroup$ The "density one" statement for Goldbach's conjecture is not difficult to prove. It is already known that the set of exceptions of Goldbach's conjecture, i.e., the set of even numbers which cannot be written as a sum of two primes, has density $O(x^\delta)$ for some $0 \leq \delta < 1$ (the record is due to Montgomery and Vaughan, as far as I know). $\endgroup$– Stanley Yao XiaoCommented Dec 20, 2015 at 15:30
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$\begingroup$ I know, but I'm interested in density one for even integers that are at the same time Goldbach numbers (i.e. that can be written as the sum of two primes) and prime gaps. $\endgroup$– Sylvain JULIENCommented Dec 20, 2015 at 15:33
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3 Answers
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Goldbach's Conjecture has been verified for all numbers up to $4 \times 10^{18}$ by Oliveira e Silva, Herzog, Pardi. On the other hand, Thomas R. Nicely has shown that all even numbers up to $2000$ occur as prime gaps. This includes a list of all first known occurrences as part of an exhaustive search of primes up to $5 \times 10^{16}$.
Answer. It includes all even numbers in $[4,2000]$.
answered Dec 20, 2015 at 18:57
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Definition: A weak prime gap is an integer which occurs infinitely often as the difference between two (not necessarily adjacent) primes.
We'll show that the set of weak prime gaps has positive density, and we already know that the set of Goldbach numbers as asymptotic density $1$, which means that their intersection must also have positive density.
In particular, we use the following 'bounded gaps between primes' results:
Let $T$ be an admissible set of integers (i.e. one that misses at least one residue class modulo any prime) of cardinality $|T| \geq 50$. Then there exists some $d \in T - T$ (the difference-set of $T$) such that $d$ is a weak prime gap.
There exists an admissible set $T$ of size $50$ such that $\max(T) - \min(T) = 246$.
Now, we note that for any $k \in \mathbb{N}$, the set $kT$ is also admissible, and its difference-set only contains integers divisible by $k$ and no greater than $246k$. Consequently, we obtain the following result:
- For any $k$, at least one of the integers $k, 2k, \dots, 246k$ is a weak prime gap.
That guarantees that weak prime gaps have asymptotic density at least $\dfrac{1}{246^2}$. I don't know whether this argument can be refined to show that strong prime gaps (where the primes are required to be adjacent) also have positive asymptotic density.
answered Dec 20, 2015 at 19:58
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Note that Nicely's website lists far more prime gaps than only the numbers 4 to 2000 that Tony Huynh mentioned, but in separate tables: Tables of prime gaps (Wayback Machine)
He doesn't seem to mention up to which number all even gaps are known to occur; from manual inspection it seems that all even numbers up to 52,400 occur as gaps between two consecutive proven primes. (52,402 is only listed as the gap between two probable primes, and so are several later entries). Above 100,000, there are several even numbers which aren't even listed as gaps between probable primes, e.g. 116,254.
