Prime-Integer-Prime arithmetic progression

Proposition:

Prove (or disprove) that for every integer $n \ge 4$, there exists at least one ordered triplet $ (p_1 , n , p_2) $ where $p_1$, $n$ and $p_2$ are in arithmetic progression and $p_1$ and $p_2$ are distinct primes.

This problem came to me upon pondering over numbers. I do not know if this idea has already been discussed by an individual or any mathematical community.

#NumberTheory

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Since, $p_{1}, n, p_{2}$ are in an AP;
Therefore;
$2n=p_{1} + p_{2}$
Now, if Goldbach's conjecture is true; our proof will be complete.
There exists an AP for n=2 and 3 also because there is no restriction that tells that $p_{1}, p_{2}$ are distinct; or the AP is non-constant.

Is there any conjecture for every even integer greater than 6 as sum of 2 distinct primes?

Yatin Khanna - 4 years, 2 months ago

Yes. I've changed my problem statement to reflect the existence of distinct $p_1$ and $p_2$ .

Is there any conjecture for every even integer greater than 6 as sum of 2 distinct primes?

This is the real deal. For now, even I have only observed this result but cannot prove it.

Tapas Mazumdar - 4 years, 2 months ago

Hey try googling the prime distributions and something on Green-Tao Theorem( on arithmetic progressions of primes).See if the first is relevant to the proposition

Spandan Senapati - 4 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$