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Math Help - Number problem

  1. #1
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    Number problem

    P is the product and S the sum of two natural numbers different from zero, prove that there are infinite natural numbers that can NOT be obtained with P + S.

    Example

    5 can be obtained because: (1 * 2) + (1 +2) = 5

    Not 6 .. (can test)

    The problem is to show that there are infinite numbers such as 6.
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  2. #2
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    Quote Originally Posted by streethot View Post
    P is the product and S the sum of two natural numbers different from zero, prove that there are infinite natural numbers that can NOT be obtained with P + S.

    Example

    5 can be obtained because: (1 * 2) + (1 +2) = 5

    Not 6 .. (can test)

    The problem is to show that there are infinite numbers such as 6.

    Let p be a natural number for which there exist two natural numbers s.t.

    p = nm + n + m = (n+1)(m+1) - 1 ==> p - 1 = (n+1)(m+1).

    If every natural number p can be obtained, then since every natural number can be written as p-1 we get that every natural number can be expressed as (n+1)(m+1)...but there are infinite very easy to find natural numbers that can NOT be expressed this way...

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Let p be a natural number for which there exist two natural numbers s.t.

    p = nm + n + m = (n+1)(m+1) - 1 ==> p - 1 = (n+1)(m+1).

    If every natural number p can be obtained, then since every natural number can be written as p-1 we get that every natural number can be expressed as (n+1)(m+1)...but there are infinite very easy to find natural numbers that can NOT be expressed this way...

    Tonio
    I think Tonio meant p + 1 = (n+1)(m+1)
    Last edited by aman_cc; October 14th 2009 at 06:00 AM. Reason: Incorrect argument in my post so deleted it
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  4. #4
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    Quote Originally Posted by aman_cc View Post
    I think Tonio meant p + 1 = (n+1)(m+1)
    Of course. Thanx

    Tonio
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  5. #5
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    Quote Originally Posted by tonio View Post
    Of course. Thanx

    Tonio
    @Tonio - Hi - I was trying this question as well. I rested it on the argument that if N can be expressed as mn+m+n => N+1 can't be expressed as such.

    Is this true? (Initially it looked trivial to me, but am lost now)
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  6. #6
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    Sorry I was completely wrong. Please ignore.
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