A classic problem is to find all rectangles with integer sides such that the numeric values for the area and perimeter are equal.
There are two distinct solutions to this problem. Can you find them, with proof?
A possible variant to the classic problem is to generalize the sides into a pair of arbitrary non-zero
Gaussian integers.
That is, what pairs of non-zero Gaussian integers x and y satisfy x*y=2*(x+y)?
Since positive and negative don't have much meaning with gaussian integers, I'll consider solutions with negative reals as well.
Part 1:
We seek xy=2(x+y) first over the reals. The can be rewritten as the rational function
y=2x/(x-2)
with asymptotes x=2 and y=2.
The obvious solutions are the vertices (0,0) and (4,4) any others must be headed towards the asymptotes so we only need to check x=3 and x=-1. Both work, giving (3,6), (1,-2) [and reversals]
Part 2:
We seek numbers (a+bi) and (c+di) that give integer solutions to
ac-bd=2(a+c)
and
ad+bc=2(b+d)
I have so far found
(2+1i)(2-4i) = 8-6i
(2+4i)(2-1i) = 8+6i
(2+2i)(2-2i) = 8
but there may be others
Edit for an error in the real pair.
Edited on December 23, 2022, 2:56 pm
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Posted by Jer
on 2022-12-23 09:48:13 |
Part 1 and part of part 2
