Definition
Plain language
Two numbers whose powers never line up exactly — like 3 and 4.
As stated in the literature
A property of integers whose logarithm ratio is irrational, ensuring their integer powers can come arbitrarily close to each other but never coincide; central to the Erdős #125 proof's thinning argument.
Why it matters: This property is what lets number theorists rule out exact coincidences and run delicate counting arguments.
For example, no power of 3 is ever exactly equal to a power of 4, even though their values can come close.
Heard on the show
“… Which means powers of three and powers of four are what mathematicians call multiplicatively independent — you can find arbitrarily large integers k and m where three to the k and four to the …”Episode 067 — An AI Just Solved a 1996 Erdős Problem—and the Simplest Agent Won