A set of distinct numbers fulfiling a property > If $x, y, z, u \in \Bbb{N}$ are distinct numbers, do any set of numbers exist to fulfil the following property $$x^y =y^z =z^u.$$ > > It is certainly true for only three numbers $x, y, z$. Is there any formula to find a set of these numbers?

Not a complete answer, but here is some python code that can generate some of these (note: not even close to optimized):


from itertools import product
import math

max = 260
xyz = list(product(range(2, max), repeat=3))
for (x, y, z) in xyz:
if x != y and y != z and x != z:
if x ** y == y ** z:
u = (z * math.log(y, z))
if u.is_integer():
u = int(u)
print("x={}, y={}, z={}, u={} --> {}^{} = {}^{} = {}^{}".format(x, y, z, u, x, y, y, z, z, u))


$$ 2^{16}=16^4=4^8, \\\ 64^{8}=8^{16}=16^{12}, \\\ 256^2=2^{16}=16^{4}, \\\ 256^4=4^{16}=16^8 $$