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Math Help - 7-5=2=5-3

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    7-5=2=5-3

    3,5,7 are primes.

    And:

    7-5=2=5-3

    Question:

    Are there any triples of primes so the difference between any two of sequential number equals 2? (Except the above...)
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  2. #2
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    3,5,7 are primes.

    And:

    7-5=2=5-3

    Question:

    Are there any triples of primes so the difference between any two of sequential number equals 2? (Except the above...)
    21-19=2=19-17

    Edit : wrong , but i can't delete this (there need's to stay dumb things i'm writing so everyone can laugh) but now i'm thinking there is non another sequence like that ...
    Last edited by yeKciM; August 6th 2010 at 02:17 PM.
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  3. #3
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    Hello, Also sprach Zarathustra!

    3,5,7 are primes.

    And: . 7-5\:=\:2\:=\:5-3

    Are there any other triples of primes that differ by 2?

    Since 2 is the only even prime,
    . . any such triple of primes must be consecutive odd numbers.

    It can be shown that, in a set of three consecutive odd numbers,
    . . one of them will be divisible by 3.


    Crank out a list of three consecutive odd numbers:

    . . \begin{array}{c}\rlap{/}1-3-5 \\ 3-5-7 \\ 5-7-\rlap{/}9 \\ 7-\rlap{/}9-11 \\ \rlap{/}9-11-13 \\ 11-13-\rlap{//}15 \\ 13-\rlap{//}15-17 \\ \vdots \end{array}


    And we see that 3-5-7 is the only such triple of primes.

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  4. #4
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    3,5,7 are primes.

    And:

    7-5=2=5-3

    Question:

    Are there any triples of primes so the difference between any two of sequential number equals 2? (Except the above...)
    No because for {p, p+2, p+4}, one of them must be divisible by 3, obviously p=2 does not work, and for p>3 divisibility by 3 will mean composite.

    Edit: Soroban beat me to it!
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    1 point to Soroban and 1 point to undefined.
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  6. #6
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    Twin primes

    Just to add my two cents worth in, there may be an infinity of twin primes however (Calvin Clawson's [I]Math Mysteries[I] is worthwhile reading).
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  7. #7
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    251-257-263-269

    4 consecutive primes, any 2 consecutive have same difference (6).
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  8. #8
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Wilmer View Post
    251-257-263-269

    4 consecutive primes, any 2 consecutive have same difference (6).
    Along those lines,

    Green-Tao Theorem
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  9. #9
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    Quote Originally Posted by undefined View Post
    Along those lines,
    Green-Tao Theorem
    YIKES!! They're up to 26 consecutives; bunch of nuts !!
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  10. #10
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Wilmer View Post
    YIKES!! They're up to 26 consecutives; bunch of nuts !!
    More nutjob info here

    Primes in arithmetic progression - Wikipedia, the free encyclopedia
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