[Math] Is every odd positive integer of the form $P_{n+m}-P_n-P_m$

analytic-number-theorynt.number-theoryprime numbersreference-request

I am looking for a comment, reference, remark, or proof of three conjectures as follows:

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.

My calculations the conjecture 1 true for $x=1, 3,\dotsc, 10^7-1$ and $x = 5.4371349×10^7$, $\cdots, 5.4375349×10^7$

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

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)

Or simpler:

Conjecture 2: Is every odd positive integer of the form $P_{c}-P_a-P_b$ ?

See also:

A stronger comjecture 2

Conjecture 3 (Maillet's conjecture) Is every even positive integer of the form $P_{a}-P_b$

I have just computed the conjecture 3 is true with $x=2, 4, \cdots, 10^6$ and $3873$ numbers $x = $$9.82197492×10^8$$,$ $\cdots$$,$ $9.82226054×10^8$

Example:

$2=5-3$;

$4=7-3$;

$6=13-7$;

$8=13-5$;

$10=17-7$

See also:

Best Answer

The conjecture 3 is mentioned here and here, 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 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.

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