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Math Help - A number theoritic problem

  1. #1
    Junior Member Sarasij's Avatar
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    Smile A number theoritic problem

    Please help me with a solution to the following problem,I've been trying this for somedays but is unable to approach it in any way...
    N1=2, N2=3, N3=5, N4=6 , ... , Ni=i-th non square integer.

    It is found that for some integer m, m^2< Nn <(m+1)^2

    Prove that m= [√n + (1/2)] where [x]=Greatest Integer Function.

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  2. #2
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    Quote Originally Posted by Sarasij View Post
    N1=2, N2=3, N3=5, N4=6 , ... , Ni=i-th non square integer.

    It is found that for some integer m, m^2< Nn <(m+1)^2

    Prove that m= [√n + (1/2)] where [x]=Greatest Integer Function.


    The number of squares between 2 and N_n is  k_n:=[\sqrt{n}]-1 (why?) , so N_n=n+k_n ,

    and thus m^2<N_n<(m+1)^2\Longrightarrow ...(complete)

    Tonio
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by tonio View Post
    The number of squares between 2 and N_n is  k_n:=[\sqrt{n}]-1 (why?) , so N_n=n+k_n ,

    and thus m^2<N_n<(m+1)^2\Longrightarrow ...(complete)

    Tonio
    Perhaps it is  N_n=n+k_n+1 instead  N_n=n+k_n ?

    If not, can you show some calculations please?
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Perhaps it is  N_n=n+k_n+1 instead  N_n=n+k_n ?

    If not, can you show some calculations please?

    Of course, it is N_n=n+k_n+1 . Thanx

    Tonio
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  5. #5
    Junior Member Sarasij's Avatar
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    Re: A number theoritic problem

    Thanks a lot guys...I've solved it in another way 2...
    Nn=n+1 , 1<= n <=2
    =n+2 , 3<= n <=6
    =n+3 , 7<= n <=12...
    =n+k , k^2 - k + 1 <= n <= k^2 + k

    From here a little simplification tells that "(√n + 1/2)-1 < m < √n + (1/2)"
    which clearly implies the proof...
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  6. #6
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    Re: A number theoritic problem

    Quote Originally Posted by Sarasij View Post
    Thanks a lot guys...I've solved it in another way 2...


    No, you didn't. The following is exacty the same formula shown to you above.

    Tonio




    Nn=n+1 , 1<= n <=2
    =n+2 , 3<= n <=6
    =n+3 , 7<= n <=12...
    =n+k , k^2 - k + 1 <= n <= k^2 + k

    From here a little simplification tells that "(√n + 1/2)-1 < m < √n + (1/2)"
    which clearly implies the proof...
    .
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  7. #7
    Junior Member Sarasij's Avatar
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    Re: A number theoritic problem

    I dont get it. u are dealing with no of squares. but i investigated the range of n for the condition k2<n<(k+1)2 to hold true. plz explain.
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  8. #8
    Junior Member Sarasij's Avatar
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    Re: A number theoritic problem

    okk... the k factor in my calculations involves the k_n you have shown. yours is more generalised. can u show the proof of the statement that k_n=[√n]-1?
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    Re: A number theoritic problem

    Quote Originally Posted by Sarasij View Post
    okk... the k factor in my calculations involves the k_n you have shown. yours is more generalised. can u show the proof of the statement that k_n=[√n]-1?


    Well, I think that is pretty straightforward and I leave the easy proof to you. Please note that the -1 is there because we don't count 1...

    Tonio
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