Bleem (thank you!) definitely exists as an idea and definitely does not exist as a physical entity. But does it exist as a mathematical object?
No - I checked.
Not yet. (The complex numbers did not exist as mathemadical objects until relatively recently.)
While "checking" it you have interacted with it, so it seems you have confirmed its existence
By the way, many things that "exist as an idea" do exist within the nature itself, like the idea of gravity obviously affects the direction of the water flow in a waterfall.