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
………………………………………………..

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
………………………………………………..

 
(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
………………………………………………..

 
there are infinitely many possibilities.

 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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