The problem is somewhat tricky. I would like to generate a set of number pairs $(x_1, y_1), (x_2, y_2)...$ so that each pair of numbers has uniqueness on all four computations (addition, subtraction, multiplication, and division).
For example, $x_1 = 2, y_1 = 2$ is definitely not a good choice here because $2 + 2 = 2 * 2. x_1 = 2, y_1 = 1$, and $x_2 = 1, y_2 = 2$ are also not good choices because $x_1 + y_1 = x_2 + y_2$.
For a better explanation of my question, please see the below picture. In short, every single cell value in the red box should be unique. For example, if I get the $v_1$. then I know it is and can only be from row $1$.
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Hope my explanation is clear. Could anyone please provide me with guidance on how to generate a set of numbers like this or directions on where I should look into?
I'm not sure what tag should I use it with, please bear with me if I used any wrong tag.
Let a = 1 b > 1 ; and a,b are natural numbers
a + b = b + 1 ≠ a + bn for n > 1
a - b = 1-b ≠ 1 -bn for n > 1
a*b = b ≠ a * (bn) for n > 1
a/b = 1/b ≠ 1/(bn) for n > 1
This suggests the sequence of pairs:
$a_n$ = (1,y), y > 1
$a_{n+1}$ = (1,2y)
So, we consider:
(1,2),(1,4),(1,8),(1,16).... etc
Note: As this seems to be an Excel application.
There may be rounding errors when trying to divide 1/x; which could result in duplicate terms. Double check your output to make sure everything is as desired.