That is, for some fixed $k\in N$, what is the set of positive integers $n$ such that $(6k+2)^n+3$ and $(6k+2)^n+5$ are both prime? At least, is this set finite?
What is the set of positive integers $n$ such that $(6k+2)^n + 3$ and $(6k+2)^n + 5$ are both prime?
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6If we knew it was infinite, then the twin primes conjecture would be solved. So the only possible answers (currently) regarding its size are "finite" or "unknown". – Zev Chonoles Jun 30 '13 at 11:33
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As $2^n=(3-1)^n\equiv1\pmod 3$ if $n$ is even, then $(6k+2)^n+5\equiv 0\pmod 3,$ hence composite ,so $n$ must be odd – lab bhattacharjee Jun 30 '13 at 11:33
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If $n\equiv 1\pmod 4$ then $2^n\equiv 2 \pmod 5$. So, $n=1$ or $n\equiv 3\pmod 4$. Similarly, $n\not\equiv 1\pmod 6$ if $n>1$. – Thomas Andrews Jun 30 '13 at 11:35
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Let's pretend that the twin prime conjecture is true. As such, I am interested in finding the exact set for $n$, or at least if there is a proof that it is for sure finite. – Michael Jun 30 '13 at 11:42
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2This will not help you in this case. If we pretented it were wrong, then you can deduce, that this set is finite, not conversely. – Tomas Jun 30 '13 at 11:43
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Still, the only answer you are likely to be able to get is negative. Knowing the TPC doesn't really help with this problem. – Thomas Andrews Jun 30 '13 at 11:43
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I added the "for some $k$" in your title, to reflect what you wrote . – Calvin Lin Jun 30 '13 at 14:29
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1The case where $6k+2$ happens to be a power of two, in particular the cases $k=0$ and $k=1$, are covered by this question. – Jyrki Lahtonen Jun 30 '13 at 19:21