## (x + a/b)(y – c/d) = x*y

$(43 \; + \; 2/5) \,(31 \; - \; 2/7) \; = \; 43\times 31$

$(341 \; + \; 2/3) \,(205 \; - \; 2/5) \; = \; 341\times 205$

$(57 \; + \; 1/3) \,(43 \; - \; 1/4) \; = \; 57\times 43$

$(781 \; + \; 1/2) \,(521 \; - \; 1/3) \; = \; 781\times 521$

Find an infinite set of products having the same property.

math grad - Interest: Number theory
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### 4 Responses to (x + a/b)(y – c/d) = x*y

1. paul says:

I found 427 such sets with 1<= a, b, c, d <=9.

Each set can be produced with a sequence function for x and y and a corresponding one for (x * y).

For example with a/b = 1/2 and c/d = 1/3 the sequence functions are
3 n-2 , 2 n-1 , 6 n^2-7 n+2 for (x, y and x * y respectively).

The 427 set sequence functions are here

https://www.dropbox.com/s/ziuk1rncb3eiolc/x%20plus%20a%20over%20b.txt?dl=0

Paul.

2. paul says:

The above example produces the following with n = 1 to 10.

(1 + 1/2)(1 – 1/3) = 1 x 1
(4 + 1/2)(3 – 1/3) = 4 x 3
(7 + 1/2)(5 – 1/3) = 7 x 5
(10 + 1/2)(7 – 1/3) = 10 x 7
(13 + 1/2)(9 – 1/3) = 13 x 9
(16 + 1/2)(11 – 1/3) = 16 x 11
(19 + 1/2)(13 – 1/3) = 19 x 13
(22 + 1/2)(15 – 1/3) = 22 x 15
(25 + 1/2)(17 – 1/3) = 25 x 17
(28 + 1/2)(19 – 1/3) = 28 x 19

-2+3 n, -1+2 n, 2-7 n+6 n^2

Paul.

• benvitalis says:

You could write (28 + 1/a)(19 – 1/b) = 28*19
a = 19*n + 2 and b = 28*n + 3

• benvitalis says:

Would a product of 3 factors be possible?
(x ± a/b) (y ± c/d) (z ± e/f) = x*y*z