This morning I’ve been playing with powers. By this I mean relationships such as 4=2² or 16=4². What particularly fascinates me are numbers that can be expressed both as a^b and b^a, where a and b are real numbers. For example, 16=4² and 2⁴. Is it possible to find all such numbers? Also, what about numbers that are, say, 20 units apart that bear the same property? Or 3,000 units apart? We can express this problem symbolically using the following relationship:
Re-arranging these variables, we can say that:
Setting a=x, b=y, and keep σ as a slider, we can plot this relationship.
Because this is kind of overwhelming, let’s focus on a single case, say σ=0:
What does this mean? Well, we’d expect to see points (a,b) corresponding with numbers we know meet the given condition. We can re-use our previous example of 16=4²=2⁴. We therefore expect the points (2,4) and (4,2) to appear on this plot. If you look closely, you can see that this is met. The truly interesting cases don’t exist on the diagonal line. These cases simply say that, for example, 6⁶=6⁶. The curve is much more interesting because it gives fascinating cases such as 2.1^(3.713)=3.713^(2.1). Also interesting is the intersection point of the curve and the diagonal. This is the only combination of “a” and “b” that doesn’t have a solution where a≠b. This point is, beautifully, (e,e), where e=2.71828…. or “Euler’s number”. Unfortunately, π does not have a beautiful combination!
We can also find combinations of numbers that are some distance apart. For example 3² and 2³ are one unit apart (9–8=1). Because we have the σ term, we can find combinations of (a,b) that are as far apart as we want. Let’s use our previous example as a trial: Which numbers are possess the property of a^b=20+b^a? To find an example of such numbers, we simply set σ=20 and plot:
Interestingly, almost all the numbers here have two cases. For example, 4.989⁵=20+5^(4.989) and 2.246⁵=20+(5^(2.246)). The only number with one case is the point at the far left of the curve. Studying general one-case solutions seems like an interesting side-project.
These are just some ideas I’ve been thinking about!
