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Math Help - diophantine equations

  1. #1
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    diophantine equations

    show that if n = ab - a - b, then there are no nonnegative solutions of ax + by = n.

    Thanks
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  2. #2
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    Quote Originally Posted by suedenation View Post
    show that if n = ab - a - b, then there are no nonnegative solutions of ax + by = n.

    Thanks
    I did not solve the problem yet, but I was able to simplify what you are saying.
    ---
    You have,
    aX+bY=ab-a-b
    Equivaleny,
    a(X+1)+b(Y+1)=ab
    Call x=X+1 and y=Y+1
    Thus,
    ax+by=ab
    Where x and y are positive integers
    Because,
    X,Y where non-negative implies X+1,Y+1 are positive.

    Thus, you need to show for a,b not equal to zero,
    The linear diophantine equation,
    ax+by=ab
    Has no positive solutions.

    Begin by noting the trivial solutions:
    (x,y)=(b,0) and (x,y)=(0,a)
    Then use them to construct your basis of solutions.
    But I did not get any further.
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  3. #3
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    Okay, I finally solved it.

    Again, using my previous post it is equivalent to saying,
    ax+by=ab
    Has no positive solutions x,y.

    But that is not true!
    10x+8y=80
    Has a solution,
    x=4 y=5

    Thus, what you need to add is that gcd(a,b)=1; a,b>0

    Then we have,
    ax+by=ab
    A trivial solution is, (but not positive)
    x=b y=0
    Thus, all solutions are, for integer t
    x=b-bt
    y=at
    We need that,
    x,y>0
    Thus,
    b-bt>0
    bt>0
    Thus,
    b>bt
    bt<0
    Thus,
    0<t<1
    Which is an impossible because t is an integer.
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