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Mar 2015
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Universe 2.71828i3.14159
Hello people. I need all none-negative integers $(a_1,a_2,b_1,b_2)$ where $$a_1+a_2=b_1+b_2 \; \\ \\ \\ a_1a_2 =b_1b_2 \; $$

$a_i \ne b_j, \; \; i,j=1,2$
 
Jun 2019
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$a_i \ne b_j, \; \; i,j=1,2$
You mean $a_1\ne b_1, a_2\ne b_1, a_1\ne b_2, a_2\ne b_2$?

I'm pretty sure there are none.
The equality of the products means we can write $a_1 = nb_1, a_2 = \frac{b_2}{n}, n\ne 1$. Plug this into the equality of the sums, and you get
$(n-1)b_1 + \left( \frac{1}{n} - 1 \right) b_2 = 0$.
With integer b values, $n=1$ is the only way this works.

Just to double-check, I tested the values for $a_1, a_2, b_1 \in \{ 0, 1, …, 10000 \}$, and there were no matches.
 
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