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Let $k$ be an odd number of the form $k=2p+1$ ,where $p$ denote any prime number, then it is true that for each number $k$ at least one of $6k-1$, $6k+1$ gives a prime number.

Can someone prove or disprove this statement?

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If this were true, don't you think that finding big prime numbers would be a whole lot easier? – orlp Sep 12 '11 at 16:40
@nightcracker,this question will be considered as rhetorical – pedja Sep 12 '11 at 16:50
up vote 20 down vote accepted

$p = 59 \implies k = 2p + 1 = 119$. Neither $6k+1 = 715$ nor $6k-1 = 713$ is prime. Some other counter examples are:

59 83 89 103 109 137 139 149 151 163 193 239 269 281

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You got confused with your quantifiers, but if your conjecture is what I guess it is, then the first five counterexamples are $p=$ 59,83,89,103,109.

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