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Math Help - prime number theorem

  1. #1
    Eater of Worlds
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    prime number theorem

    A friend of mine posed an interesting prime number theorem I have never seen before. He claims he thought this up on his own.

    Suppose you take any prime number, then double it. Say, 2p.

    Then, accordingly to his theory, you can add or subtract any integer 2 through 9 and get another prime number.

    Say we take 1009, double it and get 2018. If we add 9 we get another prime.

    Which is 2027. This is prime.

    Can we prove or disprove in the general case and find an instance where it doesn't hold?.

    Intuitively, one would think that it will break down after the numbers get so big.
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    Quote Originally Posted by galactus View Post
    A friend of mine posed an interesting prime number theorem I have never seen before. He claims he thought this up on his own.

    Suppose you take any prime number, then double it. Say, 2p.

    Then, accordingly to his theory, you can add or subtract any integer 2 through 9 and get another prime number.

    Say we take 1009, double it and get 2018. If we add 9 we get another prime.

    Which is 2027. This is prime.

    Can we prove or disprove in the general case and find an instance where it doesn't hold?.

    Intuitively, one would think that it will break down after the numbers get so big.
    Are you saying that 2p - n must be a prime for some value of n? If so there is a theorem (apparently by M. Wolf) that says we can find a consecutive list of primes to any length we desire.

    -Dan
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    Eater of Worlds
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    Are you saying that 2p - n must be a prime for some value of n?
    -Dan

    Yes, I reckon I am TQ. Thanks, I will try to find that. I was not familiar with such a theorem, but then again, there are oodles of prime number theorems.
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    I have snooped around, but can't find anything regarding Wolf you mentioned.

    What I was shown and asked about was that if you take any prime number,

    say 11, Then double it, You get 22. Now add or subtract any integer between 2 and 9 and you will get another prime.

    If we add 7 to 22, we get 29. If we subtract 2, we get 19. And so on.

    These are small numbers though. I would think that when you get up to large numbers it probably won't hold because the distribution gets thinner compared to small numbers like 2 through 9.

    The person who showed me this said he had checked them up to the first 1000 primes and it held. He was wondering when it failed. An example of a prime that when doubled, then add or subtract any number 2 through 9 you won't get another prime.

    Let's try a bigger one. Take 100213

    Double it and get 200426

    Now, if we add or subtract some integer between 2 and 9 let's see if we get another prime.

    Because 200246-9 does result in a prime.

    Let's try something bigger. Say, 125987791. Double it and get 251975582

    Now, let's see if it holds.

    Yep, it holds. 251975582-3 results in another prime.

    I thought this was a cool thing to think up on ones own, so I figured I would check into it by asking some more learned than I on prime number distributions.

    Does anyone know of a way we can prove or disprove this, generally?.

    And/or give an example where it does not hold.
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  5. #5
    Moo
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    Hello,

    Maybe i misunderstood the term "integer", in which case, excuse me...

    11 -> 22 -> 30 ?

    Sorry, did you mean "one integer between 2 and 9 will make it prime" ?
    Last edited by Moo; March 11th 2008 at 03:51 PM.
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    Moo
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    say 11, Then double it, You get 22. Now add or subtract any integer between 2 and 9 and you will get another prime.

    If we add 7 to 22, we get 29. If we subtract 2, we get 19. And so on.
    If we substract 2, we get 20, which is obviously not prime...

    Something is disturbing me in it... If you say that any integer between 2 and 9 works, then, it's false, because any even integer added or substracted to 2p will equal a non-prime number (because even-even=even)...
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    Quote Originally Posted by galactus View Post
    Yes, I reckon I am TQ. Thanks, I will try to find that. I was not familiar with such a theorem, but then again, there are oodles of prime number theorems.
    Wolf might not be the right name. (It was a quick reference I saw when I did a net search.) However there is some sort of theorem about the frequency of primes decreasing (on average) as the numbers climb. I wish I could help you more by giving a source or name.

    -Dan
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    A quick trawl through a table of primes shows that 673 is a prime. Double it and you get 1346. But the table shows that there are no primes between 1327 and 1361.
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    I did not except this conjecture to hold up. The Distribution of primes is more complicated then being broken up into an interval from -9 to +9.
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  10. #10
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    Me neither, PH. I was trying to find an example but couldn't spot it until Opalq pointed one out.
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