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Math Help - Sequence of natural numbers

  1. #1
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    Sequence of natural numbers

    The sequence of natural numbers satisfies for all . Prove that

    Can anyone show me a full solution to this question? (I'm quite noob at number theory since I just started it, please don't leave out any steps!!)
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  2. #2
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    Quote Originally Posted by usagi_killer View Post
    The sequence of natural numbers satisfies for all . Prove that

    Can anyone show me a full solution to this question? (I'm quite noob at number theory since I just started it, please don't leave out any steps!!)


    What does above mean plz? Is it a std notation?
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  3. #3
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    Quote Originally Posted by usagi_killer View Post
    The sequence of natural numbers satisfies for all . Prove that

    Can anyone show me a full solution to this question? (I'm quite noob at number theory since I just started it, please don't leave out any steps!!)
    The number 1 has the property that GCD(1,j) = 1 for all j, and it is the only number with that property. Therefore a_1=1.

    The number 2 has the property that GCD(2,j) = 1 or 2 for all j, and it is the only number with that property. Therefore a_2=2.

    Can you continue that line of reasoning?
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  4. #4
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    Thanks for that, but I have no idea how to complete the proof... could you please show me?

    thanks again.
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  5. #5
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    Use proof by induction.
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  6. #6
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    Quote Originally Posted by Opalg View Post
    The number 1 has the property that GCD(1,j) = 1 for all j, and it is the only number with that property. Therefore a_1=1.
    I'm probably missing something here, but does that not only tell you that a_1 is coprime to all the other terms in the sequence and is the only term in the sequence with this property?

    CB
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  7. #7
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    Quote Originally Posted by CaptainBlack View Post
    I'm probably missing something here, but does that not only tell you that a_1 is coprime to all the other terms in the sequence and is the only term in the sequence with this property
    No, it was me missing something (misreading the question, as so often). I was thinking that the sequence was meant to be a permutation of the natural numbers. But the question does not say that, so it is harder than I first thought. I doubt whether I'll have time to reconsider it on Christmas Day however.
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  8. #8
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    I think I got it:

    First assume

    Therefore for some , let's call it ,



    for some



    Now





    But Contradiction!

    Thus so
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