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Math Help - A conjecture involving primes

  1. #1
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    A conjecture involving primes

    Hi everybody,
    Does anyone have more info (or a counter example) about:

    Any odd prime P can be written in form:
    <br />
P= p+2^a<br />

    where p is prime and a>=1

    In fact i think that every odd number can be written in the above form

    Thank you a lot.
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  2. #2
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    Well, there's a slight problem ... 3 = 2 + 1, and I'm assuming that we're not considering 1 to be prime ... so then a = 0 (a is alpha ... too early for me to be using latex, hehehe). So you should allow for a = 0 .

    Just some other things, for odd primes, it's obvious that P = p + 2n ... i.e. the difference between any odd primes is even .

    As for P = p + 2^a, we might be able to reconstruct P so that since P = p_1 + 2n_1 where p_1 is the next prime before P, then P = (p_2 +2n_2) + 2n_1 where p_2 is the next prime before p_1, so we keep doing this until we get P = p_i + 2(n_i+...+ n_2+n_1) where \sum_{j=1}^i n_j = 2^\beta ... and I guess it's up to you to show that it's possible ... or not

    Also, not quite the same, but you might want to look up the twin primes conjecture (and related/modified conjectures), it might give you some ideas ....
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  3. #3
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    Ah ... here's another idea ... instead of breaking it down, you can try to build up ...

    like consider all primes of the form 3 + 2^a, 5 + 2^a, ... , p + 2^a, ... and check if that covers all odd primes (maybe you could show that it covers all odd numbers? so all odd numbers can can be written in that form, and all odd numbers includes all odd primes ....)
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  4. #4
    MHF Contributor chisigma's Avatar
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    Counterexample: P=97 [prime...], a=6 [>1...],  \rightarrow p=97 -2^{6} = 97-64=33=11\cdot 3 [not a prime...]

    Kind regards

    \chi \sigma
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  5. #5
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    Quote Originally Posted by chisigma View Post
    Counterexample: P=97 [prime...], a=6 [>1...],  \rightarrow p=97 -2^{6} = 97-64=33=11\cdot 3 [not a prime...]

    Kind regards

    \chi \sigma
    Yes but still
    <br />
89+2^3=97<br />
    89 is prime
    Last edited by gdmath; October 21st 2009 at 02:21 AM.
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    gdmath ... I just got another idea for the second method.

    Firstly, aside from the case of P=3, I think p is also an odd prime (since if p=2, and we add 2^a, then we just get even numbers)

    So, we start with when p=3, and we consider the set of all odd integers of the form 3+2^a. This is {5,7,11,19,35,...}. We can see that the gap gets bigger as a increases (I get the feeling that it would also help to analyze this as a sequence/series).
    Then, we consider the next prime, when p=5. 5+2^a will give us {7,9,13,21,37,...}. We keep doing this, and what you will notice is that we fill up the gaps ... so if any odd number O can be written in the form O = p + 2^a, then any odd prime can be written in this form .
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  7. #7
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    Opalg's Avatar
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    127 looks like a counterexample.
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  8. #8
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    Quote Originally Posted by Opalg View Post
    127 looks like a counterexample.
    Indeed...

    However in a not tight version of the initial definition:
    <br />
2^{44}-127 = 17592186044287<br />
    where 17592186044287 is prime.

    Anyway thank you, your note reveals that i have more work to do on the initial assumtion
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  9. #9
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by gdmath View Post
    Indeed...

    However in a not tight version of the initial definition:
    <br />
2^{44}-127 = 17592186044287<br />
    where 17592186044287 is prime.

    Anyway thank you, your note reveals that i have more work to do on the initial assumtion
    Why are you working towards conjecturing something? In the way I see it, conjectures are usually stumbled upon, not sought. A famous quote of Gauss comes to mind...
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  10. #10
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    I do not think that i understand the purpose of your scoffing quote.

    Anyway - since you mensioned it - I work on arbitary arithmetic patterns (for computer analysys). From there i come, ussualy upon some curious relations and i want to know if they exist or no.

    As far it concern Gauss ... i do not know what of his quotes you have in mind but forgive me i do not care to know.
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  11. #11
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by gdmath View Post
    I do not think that i understand the purpose of your scoffing quote.

    Anyway - since you mensioned it - I work on arbitary arithmetic patterns (for computer analysys). From there i come, ussualy upon some curious relations and i want to know if they exist or no.

    As far it concern Gauss ... i do not know what of his quotes you have in mind but forgive me i do not care to know.
    Sorry, I didn't mean to offend. However it seems that you posted this conjecture because you didn't feel like verifying it by yourself, preferring to leave the work up to somebody else. (By "verification", I mean checking more than the primes less than 150, of which there are only 35.) As it turns out, you wouldn't have needed to search very far to realize that the conjecture is false.

    By no means do I want to discourage you from searching for mathematical truth. But there are better ways than to stumble into a dark alley, to pick up an old hat that's lying there, conjecture that it's a cat and send it to a laboratory for analysis to make sure that it is. Simply walking out of the alley and into the dimmest of streetlights will reveal its true nature.
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  12. #12
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    Sorry for me also if iwas offensive.

    You also are partially right, I should be more carefull for the threads I post.

    Thank you
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  13. #13
    MHF Contributor Bruno J.'s Avatar
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    We're all good my friend! Good luck in your endeavours.
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