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Math Help - Divisibility theory in the integers

  1. #1
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    Divisibility theory in the integers

    Prove the following concerning triangular numbers:
    a) A number is triangular if and only if it is of the form n(n+1)/2 for some n≧1
    b) the integer n is a triangular number if and only if 8n+1is a perfect square
    c) the sum of any two consecutive triangular numbers is a perfect square
    d) if n is a triangular, then so are 9n+1, 25n+3,and 49n+6
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  2. #2
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    Hello, harrietchemist!

    I can help with the last two . . .


    Prove the following concerning triangular numbers:

    c) the sum of any two consecutive triangular numbers is a perfect square
    Let the first triangular number be: .T1 .= .k(k+1)/2
    then the next triangular number is: .T
    2 .= .(k+1)(k+2)/2

    Then: .T
    1 + T2 .= .k(k+1)/2 + (k+1)(k+2)/2 .= .(k + k + k + 3k + 2)/2

    . . = .(2k + 4k + 2)/2 .= .k + 2k + 1 .= .(k + 1) . . . a square



    d) if n is a triangular, then so are: 9n + 1, 25n + 3, and 49n + 6
    Since n = k(k+1)/2, then: .9n + 1 .= .9k(k+1)/2 + 1

    . . = .(9k + 9k + 2)/2 .= .(3k + 1)(3k + 2)/2 . . . a triangular number

    Do the same for the other two expressions.

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  3. #3
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    Quote Originally Posted by harrietchemist View Post
    Prove the following concerning triangular numbers:
    a) A number is triangular if and only if it is of the form n(n+1)/2 for some n≧1
    b) the integer n is a triangular number if and only if 8n+1is a perfect square
    Hmm, these problems seem to be taken from:
    Burton, David. Elementary Number Theory 5/e. Page 15

    Anyway.

    a)Triangle means, t_n = 1+2+...+n
    Use the summation formula 1+2+...+n=n(n+1)/2

    b)If t_n is trianglular then it has form n(n+1)/2
    Then, 8*t_n+1=8n(n+1)/2 + 1 = 4n(n+1)+1 = 4n^2+4n+1=(2n+1)^2

    You try to do the converse.
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  4. #4
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    thank you very much ~
    I study number theory by myself...so I really happy that you both help me...
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