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Math Help - Prime numbers

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    Prime numbers

    If the numbers p, p+2 and p+4 are primes, find p.
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  2. #2
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    Hello,
    Quote Originally Posted by Apprentice123 View Post
    If the numbers p, p+2 and p+4 are primes, find p.
    - Any integer can be written in the form 3k,3k+1 or 3k+2, for some integer k \geq 0
    - Among these three possibilities, only 3k+1 and 3k+2 can be primes and 3 is a prime.

    Now if p=3k+1
    {\color{red}p+2=3k+3=3(k+1)}
    p+4=3k+4
    p+2 is prime if and only if k+1=1, that is k=0. Otherwise, it is a multiple of 3 and hence is not prime.
    But p=3x0+1=1 is not a prime.
    Hence no prime number in the form 3k+1 satisfy the condition.


    Now if p=3k+2
    p+2=3k+4
    {\color{red}p+4=3k+6=3(k+2)}
    p+4 is prime if and only if k+2=1, that is k=-1.
    This gives p=-3, p+2=-2, p+4=1.
    These are obviously not prime numbers. (they have to be positive).


    Now if p=3, p+2=5, p+4=7
    They're all prime numbers.
    And it's the only possibility for p.
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  3. #3
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    Thank you very much, the exercise was not easy.
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