## Products | (x + r_1)(y + r_2) = xy; (r_1, r_2) are rational

$(x \; + \; r_1)(y \; + \; r_2) = x y$

Example #1 :    $r_1 = -1/2$,    $r_2 = 1/3$

$(x \; - \; 1/2)(y \; + \; 1/3) = x y$

Integer solutions:    $x = 3 n+2$,    $y = 2 n+1$

$(5 \; - \; 1/2)(3 \; + \; 1/3) = 15 = 5 \; \times \: 3$
$(8 \; - \; 1/2)(5 \; + \; 1/3) = 40 = 8 \; \times \: 5$
$(11 \; - \; 1/2)(7 \; + \; 1/3) = 77 = 11 \; \times \: 7$
$(14 \; - \; 1/2)(9 \; + \; 1/3) = 126 = 14 \; \times \: 9$
$(17 \; - \; 1/2)(11 \; + \; 1/3) = 187 = 17 \; \times \: 11$
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Example #2 :    $r_1 = 1/2$,    $r_2 = -1/3$

$(x \; + \; 1/2)(y \; - \; 1/3) = x y$

Integer solutions:    $x = 3 n+1$,    $y = 2 n+1$

$(4 \; + \; 1/2)(3 \; - \; 1/3) = 12 = 4 \; \times \: 3$
$(7 \; + \; 1/2)(5 \; - \; 1/3) = 35 = 7 \; \times \: 5$
$(10 \; + \; 1/2)(7 \; - \; 1/3) = 70 = 10 \; \times \: 7$
$(13 \; + \; 1/2)(9 \; - \; 1/3) = 117 = 13 \; \times \: 9$
$(16 \; + \; 1/2)(11 \; - \; 1/3) = 176 = 16 \; \times \: 11$
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$(x \; - \; 1/2)(y \; + \; 2/3) = x y$

Integer solutions:    $x = 3 n+2$,    $y = 2 (2 n+1)$

$(2 \; - \; 1/2)(2 \; + \; 2/3) = 4 = 2 \; \times \: 2 \; = \; 2^2$
$(5 \; - \; 1/2)(6 \; + \; 2/3) = 30 = 5 \; \times \: 6$
$(8 \; - \; 1/2)(10 \; + \; 2/3) = 80 = 8 \; \times \: 10$
$(11 \; - \; 1/2)(14 \; + \; 2/3) = 154 = 11 \; \times \: 14$
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there are infinitely many possibilities.