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Hi hermit;
hermit wrote:I think this means for there to be integer solution x,y to the original equation that the determinant of A must evenly divide each of the entries a, b, c, and d.
That does not hold.
The matrix eqtn.
Has a solution of x = -2 and 9 while the det of A is -11 which does not divide a or b or c or d
I see what you mean but think I misexplained the question. I will try to rephrase. What conditions must a,b,c,d satisfy so that the equation A(x,y)=(s,t) has integer solutions for all (s,t). I think what this counterexample shows is for a particular (s,t). In the question I am looking at, a,b,c,d are fixed, but s,t are not fixed integers.
I have not had time to look at link you gave yet, but will soon. Thank you again.
First, thank you very much. I thought more after seeing your reply. Here is what I have now:
Using matrix form you provided,
I used formula for inverse of 2x2 matrix and call them A and A^-1, and multiply on both sides with the inverse so that there is x,y on left, and A^-1 * s,t on right of equality. I think this means for there to be integer solution x,y to the original equation that the determinant of A must evenly divide each of the entries a, b, c, and d.
The exercise asks to find conditions on a,b,c,d so that integers x,y always can be found to satisfy original equation for any s,t. Would this condition be a sufficient answer to this question?
I have not done problem like this in a long time, but think I remember the strategy:
Fix an arbitrary point on the circle. Compute the tangent vector at this point. The magnitude of this vector is the speed. Then differentiate the tangent vector to obtain the normal vector at the point. The normal vector will always point toward the center, this should be evident in the expression for normal vector.
x(a,b) + y(c,d) = (s, t)
What conditions do (a,b) and (c,d) have to satisfy for this to have solution in integers? Of course a,b,c,d,s,t are fixed integers.
I turned this to two equations ax + cy =s, and bx + dy = t. I use y = (sc-ax)/c to substitute into second equation. This gives x = (tc-ds)/(bc-ad). Does this mean that bc-ad can not be zero? Or just that bc-ad has to divide tc-ds? Both? Do I have to give conditions in terms of ordered pairs, or just state relations regarding a,b,c,d?
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