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Math Help - Prime numbers

  1. #1
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    Prime numbers

    Try this ... from AIMO 2007 Intermediate Paper

    Find a prime, p, with the property that for some larger prime number, q, both 2q - p and 2q + p are prime numbers. Prove that there is only one such prime p.

    I don't even know where to start!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by BG5965 View Post
    Try this ... from AIMO 2007 Intermediate Paper

    Find a prime, p, with the property that for some larger prime number, q, both 2q - p and 2q + p are prime numbers. Prove that there is only one such prime p.

    I don't even know where to start!
    i'll take p=3..

    for q=5: we have 7 and 13
    for q=7: 11 and 17
    for q=13: 23 and 29
    for q=17: 31 and 37
    ....

    maybe, i'll try to come up with the proof on uniqueness soon...
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  3. #3
    Newbie Catherine Morland's Avatar
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    Any prime p > 3 is of the form 6k\pm1 for some integer k\ge1. For p > 3, you can try all combinations of p=6k_1\pm1 and q=6k_2\pm1 and verify that 2q\pm p is never prime.

    If p = 2, then 2q+p and 2q-p are both even and neither is equal to p, so neither is prime.

    Hence p = 3 is the only prime satisfying the given property.
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  4. #4
    MHF Contributor kalagota's Avatar
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    wow!

    but how to show that any prime p>3 is of the form 6k \pm 1 for some integer k\geq1?
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by kalagota View Post
    wow!

    but how to show that any prime p>3 is of the form 6k \pm 1 for some integer k\geq1?
    All odd integers are of the form 6k\pm 1 or 6k \pm 3, the latter are all divisible by 3 and so if greater than 3 itself are not prime.

    RonL
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  6. #6
    Newbie Catherine Morland's Avatar
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    Quote Originally Posted by kalagota View Post
    but how to show that any prime p>3 is of the form 6k \pm 1 for some integer k\geq1?
    If a prime is greater than 3, then it's not divisible by 2 or 3. Therefore it cannot be of the form 6k, 6k+2, 6k+3 or 6k-2, otherwise it would be divisible by 2 or 3 (or both). Hence all primes greater than 3 must be of the form 6k+1 or 6k-1.
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