# 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.

Note by Tapas Mazumdar
1 year, 4 months ago

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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?

- 1 year, 4 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.

- 1 year, 4 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

- 1 year, 4 months ago