How can I find the integer solutions for the diophantine equations $axy +bx + cy =d$ ?
the smallest particular solution ($x_0$,$y_0$) and a way to generate the rest.
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Bill Dubuque
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Loers Antario
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See completing the product (or rectangle, a generalization of completing the square. – Bill Dubuque Jun 29 '23 at 01:55
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From $axy+bx+cy=d,$ multiplying through by $a$ and factoring gives $$(ax+c)(ay+b)=ad+bc.$$ So for particular $a,b,c,d$ there can only be finitely many solutions, and an admittedly unsatisfactory way to look for them is to factor $ad+bc$ in all possible ways and set the two factors equal to the two linear terms on the left and solve, to see if $x,y$ come out integers.
EDIT: There is the possibility that the right side $ad+bc=0$, for which there may be infinitely many solutions, however this is the degenerate case of the original equation in which it represents the union of two lines, and is not an interesting case.
coffeemath
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2very clever method. How can I apply this method to find the integral points of the system of equations: $$4x+y+5z=2xyz$$ $$2x-5y+11z=0$$. Clearly, I can fix $x$ and go from there. I am trying to find a general method like yours if possible. Please react at your earliest convenience. – NumThcurious Apr 26 '20 at 12:07
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@NumThcurious: You should notify coffeemath similarly. =) – Jose Arnaldo Bebita Dris Jul 09 '21 at 08:37
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This is known as completing the product (or rectangle, a generalization of completing the square. – Bill Dubuque Jun 29 '23 at 01:55