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If p is a prime and k is a positive number less than p, and $2^k$ is incongruent to 1, $2^{(n-1)}$ is congruent to 1, then $kp^2+1$ would be a prime number?

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  • $\begingroup$ Firstly, no formula as simple as that always produces primes. Secondly, it's not clear what you mean by "congruent to 1" and "incongruent to 1" since you do not specify modulo what. Thirdly, you give no reason to believe this is true. $\endgroup$
    – Will Sawin
    May 28, 2012 at 22:17
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    $\begingroup$ He probably means congruences modulo $n$, so one of his conditions is that $n$ is a base $2$ pseudoprime, if not prime. Here is a list of all such numbers up to $10^{15}$: cecm.sfu.ca/Pseudoprimes If there is no counterexample to his conjecture on this list, then it gets interesting. $\endgroup$ May 28, 2012 at 23:13
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    $\begingroup$ Also posted to math.stackexchange, math.stackexchange.com/questions/150853/prime-of-the-form-kp21, without any mention of the post here. $\endgroup$ May 29, 2012 at 1:06
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    $\begingroup$ @Felipe, I verified that no counterexamples exist in this list. $\endgroup$ May 29, 2012 at 3:52
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    $\begingroup$ It's badly stated, unmotivated, rudely crossposted, and shows no sign of work on the part of OP --- but it is a real question! $\endgroup$ May 29, 2012 at 5:50

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I think that he omitted "modulo n". I tried to solve this problem with the fact : $ord_n{2}$ divides $n-1=kp^2$, and thought that his conjecture is true. But I couldn't proved it.

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