
> 
> I am looking for a comment, reference, remark, or proof of three conjectures as follows:
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> **Conjecture 1:** *Let $x$ be an odd positive integer. Then there exist two integers $n, m \ge 2$ so that $$x=P\_{n+m}-P\_n-P\_m,$$ where $P\_n$ is the $n^{th}$ prime.*
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My calculations the conjecture 1 true for $x=1, 3,\dotsc, 10^7-1$ and $x = 5.4371349x10^7$, $\cdots, 5.4375349x10^7$


A general of the conjecture above as follows **(but weaker)**:



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> Let every integer $r\_0$ exist positive integer $x\_0$ such that every odd number $x \ge x\_0$ has the form $x=P\_{n+m+r\_0}-P\_n-P\_m$, where $m, n \ge 2$ (PS: inspired from the comment of Lev Borisov be low)
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Or simpler:



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> > ***Conjecture 2:** Is every odd positive integer of the form $P\_{c}-P\_a-P\_b$ ?*
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**See also:**


* [Goldbach's weak conjecture](https://en.wikipedia.org/wiki/Goldbach%27s_conjecture)


**A stronger comjecture 2**



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> > **Conjecture 3 ([Maillet's conjecture](https://arxiv.org/pdf/1206.0149.pdf))** *Is every even positive integer of the form $P\_{a}-P\_b$*
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I have just computed the conjecture 3 is true with $x=2, 4, \cdots, 10^6$ and $3873$ numbers $x = $$9.82197492x10^8$$,$ $\cdots$$,$ $9.82226054x10^8$


**Example:**


$2=5-3$;


$4=7-3$;


$6=13-7$;


$8=13-5$;


$10=17-7$


**See also:**


* [Goldbach's strong conjecture](https://en.wikipedia.org/wiki/Goldbach%27s_conjecture)



Is every odd positive integer of the form $P_{n+m}-P_n-P_m$?
The **conjecture 3** is mentioned [here](https://mathoverflow.net/questions/111196/name-of-a-conjecture-on-difference-of-prime-numbers) and [here](https://arxiv.org/pdf/1206.0149.pdf), in which it is called the Maillet conjecture.


The **conjecture 2** is true. 


It can be seen by rewriting $n=P\_c−P\_a−P\_b$ as $P\_a+P\_b=P\_c-n$, and it has a solution as long as the sets $\left\{x|x=P\_a+P\_b\right\}$ and $\left\{x|x=P\_c-n\right\}$ have nonempty intersection. Call the two sets $X$ and $Y$ respectively, and let $A\_N$ denote the elements of a set $A$ which is not greater than $N$. 


We have $|X\_N|=O(\frac{N}{logN})$, and a result of [Montgomery and Vaughan](https://en.wikipedia.org/wiki/Goldbach%27s_conjecture#Rigorous_results)
 shows that $|Y\_N|≥\frac{N}{2}-CN^{1-c}$ for some $C$ and $c$.
As $|X\_N\cup Y\_N|≤\frac{N}{2}+n$ and
$|X\_N|+|Y\_N|≥\frac{N}{2}+n$ for sufficiently large $N$, it turns out that the two sets have nonempty intersection.


