# Related to the sum of consecutive primes

Yesterday I saw this question: A question about divisibility of sum of two consecutive primes (you should read the OP to understand the full problem), it just asks to prove that for all $$k\in \mathbb Z^+$$, there exist infinitely many consecutive primes such that : $$k\mid p_{n+1}+p_n.$$ the guy who asked this took care of the cases where $$k=1,2,3,4,6$$. The general case where $$k$$ is any positive integer is beyond me, but I attempted to prove the case where $$k=12$$ and I wonder it the proof is correct:

Assume that the twin prime conjecture is true, which says that there exist infinitely many consecutive primes such that $$p_{n+1}-p_n=2$$

Since any prime $$\ge 5$$ is on the form $$6k\pm 1$$ and every pair of twin primes is on the form $$(6k-1,6k+1)$$, Hence $$p_{n+1}+p_n=6k+1+6k-1=12k$$ $$12\mid p_{n+1}+p_n$$

for infinitely many consecutive primes?

Note that the twin prime conjecture also implies the case where $$k=1,2,3,4,6$$, because $$1,2,3,4,6\mid 12$$

• As you mentioned , this proof requires the truth of the twin prime conjecture, but it is a valid proof. May 26, 2021 at 12:02
• if we assume the truth of polignac's conjecture, can we do better than $k=12$? @Peter
– PNT
May 26, 2021 at 13:20
• I think we can extend this result , but I have no good approach yet. Intuitively , I would guess, that we can find infinite many pairs for every $k$. May 26, 2021 at 13:22
• that's exactly what I thought in the beginning but it turns out that you can't do any better than $12$, at least that's what I've found. @Peter
– PNT
May 26, 2021 at 13:27
• Your same argument works for $8$ too. Jul 13, 2021 at 15:33

Assuming Schinzel's hypothesis , for every positive integer $$k$$ , there are infinite many positive integers $$\ n\$$ , such that $$\ kn-1\$$ and $$\ kn+1\$$ are both prime. Those primes are obviously consecutive because they have difference $$\ 2\$$ (in fact they are twin primes). The sum is $$\ 2kn\$$ which is divisible by $$\ k\$$.

Of course, Schinzel's hypothesis is much stronger than the twin prime conjecture, but at least this is some evidence that we can find infinite many pairs for every $$\ k\$$.

• There is a gap in your proof, note that $k$ is a variable, in other words, it's not fixed, so it changes with respect to the primes $kn\pm 1$. The OP asked given a fixed $k\in \mathbb Z^+$ are there infinitely many consecutive primes s.t $k\mid p_{n+1}+p_n$ @Peter
– PNT
May 26, 2021 at 13:42
• You have not read the answer carefully. For every $k$ , we can find infinite many $n$ giving a suitable twin prime pair , if Schinzel's hypotehsis is true. May 26, 2021 at 13:43
• yeah... sorry you're right. @Peter
– PNT
May 26, 2021 at 13:45
• You've said that Schinzel's hypothesis is weaker than the twin prime conjecture, but it looks like a generalization of it? @Peter
– PNT
May 26, 2021 at 13:49
• Schinzel's hypothesis is far too strong. Dickson's conjecture suffices. Jul 13, 2021 at 15:31