Set of numbers with interesting property

Question

There is an interesting property with the set $(2,2)$ $$2 + 2 = 4$$ $$2\times2 = 4$$ $$2^2 = 4$$

I am wondering is (2,2) the only set with this property.

For example, consider the set $(1,2,3)$

$$1+2+3=6$$ $$1\times2\times3=6$$ $$1^{2^3}=1$$

so this does not work.

If we restrict the number to integers, we can see the numbers have to be the power of the same number so with simple inequality this clearly wouldn't work for any other set other than $(2,2)$, but in the case of real numbers, it seems plausible that other sets might exist.

Answer 1

The three variable case has infinite solutions in the real numbers.

Steps for the solution: Solve for z:$$z=\frac{x+y}{x\ y-1}$$

Plot (using software) the equation $$x+y+z-x^{y^z}\ =\ 0$$

Subsituting for z.

The two variable case has only one solution using analogous logic.

Answer 2

If your set has only two numbers, the only possibility is $(2,2)$. Imagine there is two numbers, the first number is the number of stars and the second number is the number of squares. We can always assume that the number of squares is bigger than the number of stars.

When you add up the two numbers, the total number of shapes is at most $2$ units. Here one unit is the number of squares.

When you multiply, the number of units is the number of stars.

If there is only $1$ star, the product will be lesser than the sum by $1$. So there cannot be $1$ star.

If there are at least $2$ stars, then the product has at least $2$ units. So the product is at least equal to the sum.

For the product to be equal to the sum, there be exactly $2$ units. So there are exactly $2$ stars and exactly $2$ squares. Therefore the only possible set is $(2, 2)$.

You have already checked that $(2, 2)$ works.