If $K$ and $S$ are prime numbers then how can I prove that for some $n$, there exist prime numbers of the form $K+2n+2$ and $S-2n$?
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2Are you restricting yourself to positive primes? What values may $n$ take? – Cameron Buie Aug 20 '13 at 13:12
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3Are you trying to prove Goldbach's conjecture by induction? – roger Aug 20 '13 at 13:16
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@GerryMyerson Actually, the way it is formulated seems to ask for some $n$ such that those two numbers are both prime. This seems to be essentially Dirichlet. – Tobias Kildetoft Aug 20 '13 at 13:18
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@Tobias, I don't think you can get there from Dirichlet. But I also think I misunderstood the original question, so I'm deleting my earlier comment. – Gerry Myerson Aug 20 '13 at 13:23
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Pretty sure you can't get there at all. Take a look at my answer and see what you think. – Cameron Buie Aug 20 '13 at 13:25
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@GerryMyerson ahh, you are right. One needs more to be able to combine arithmetic progressions like that of course. – Tobias Kildetoft Aug 20 '13 at 13:25
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1@Cameron, I suspect OP will edit the question to rule out your example. We'll see. – Gerry Myerson Aug 20 '13 at 13:26
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I have changed the title of the post to reflect the body, please see that it encompasses what you meant. – Jeel Shah Aug 20 '13 at 13:31
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@gekkostate Now the question and the title of the question are two different problems. – Aug 20 '13 at 13:41
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@Amateur Fixed! Before, it said some primes but now it says some $n$ but anyway, I just copy-pasted the body into the title so it should be the same now. – Jeel Shah Aug 20 '13 at 13:42
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1@gekkostate hahaha I hope that "copy-paste" is always an identity function so that what we copied is what we`ll paste!!! – Aug 20 '13 at 13:47
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@roger: You might just be clairvoyant. – Cameron Buie Aug 21 '13 at 14:22
1 Answers
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As written, you can't, if $n$ must be an integer.
Put $K=43$ and $S=2$. Then if you're restricting yourself to positive primes, we must put $n=0$ in order for $S-2n$ to be prime, but then $K+2n+2=45$ is not prime. If you are not restricting yourself to positive primes, then since $n=0$ doesn't work, we must take $n=2$ in order for $S-2n$ to be prime, but then $K+2n+2=49$ is not prime.
Edit: Now, if $K$ and $S$ must be odd primes, then as you've observed, an affirmative answer to this question would allow us to prove Goldbach's conjecture by induction, while a negative answer would disprove it. Consequently, I must put the answer firmly down as "unknown."
Cameron Buie
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n belongs to the set of integers and K and S are both prime numbers .Now I just want to know if there exists some n such that K+2n+2 and S-2n are both prime numbers for some primes K and S – skrstars Aug 21 '13 at 13:22
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Well, there certainly exists some $n$ such that $k+2n+2$ and $s-2n$ are both primes for some primes $k$ and $s$, e.g., $k=11$, $s=17$, $n=2$. Is that really what you are asking? – Gerry Myerson Aug 21 '13 at 13:32
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2@skrstars: That is a very different question than you asked originally. Please be precise. Your original question seems to be asking whether it is true that, given *any* primes $K$ and $S$, we can find an integer $n$ such that $K+2n+2$ and $S-2n$ are both prime. (The answer to this question is "no," as I explain above.) Now, your comment seems to be asking whether it is true that there exists some integer $n$ and some primes $K,S$ such that $K+2n+2$ and $S-2n$ are both prime. (The answer to this question is "yes," as I see that Gerry has just pointed out, but this is not very interesting.) – Cameron Buie Aug 21 '13 at 13:34