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This is a homework question of mine, I am not searching for the solution but rather what it means. It seems pretty straight forward but I am a little confused as to what the $k$ in $1 < k < n-10^6$ is supposed to be.

Here is the question: Consider the number $n = 2^{1000000000000000000000000000000000} + 1$ . Suppose that it is known that none of the numbers $1 < k < n-10^6$ divide $n$ . Does it follow that $n$ is a prime number?

Again I am not asking for solutions but rather what values $k$ may take. Thank you

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The statement $1 \lt k \lt n-10^6$ shows the range of $k$. It can range from $2$ to a million and one below $n$. You are given that no $k$ in this range divides $n$.

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    $\begingroup$ oh right of course! Thanks $\endgroup$ – user2710184 Apr 8 '14 at 2:36
  • $\begingroup$ so then we cannot say whether or not n is a prime just because the values 1 < k < n−10^6 do not divide n right? $\endgroup$ – user2710184 Apr 8 '14 at 2:38
  • $\begingroup$ Yes we can, for the $n$ under consideration. If one of those numbers divides $n$, what is the quotient? $\endgroup$ – Ross Millikan Apr 8 '14 at 2:42
  • $\begingroup$ Correct me if im wrong but n is only a prime if the values 1 < k < n/2 do not divide n right? $\endgroup$ – user2710184 Apr 8 '14 at 2:47
  • $\begingroup$ That is true. The upper limit can be reduced to $\sqrt n$, which is the point of my last comment. Can you see why? $\endgroup$ – Ross Millikan Apr 8 '14 at 3:01

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