Physical constants sometimes show unexpectedly simple mathematical relationships that may signal deeper theoretical connections. According to Amir, Lemeshko and Tokieda, the ratio of proton mass to electron mass (1836.15267245) matches 6π⁵ (1836.1181) to five significant figures[^1]. The researchers developed a probability model to determine whether such matches are coincidental or meaningful, finding that for 5-digit numbers like 1836.1, there is only a 1.2% chance of a random match with simple mathematical expressions[^1].

The study also highlights proven cases where numerical coincidences revealed real physics, such as the Stefan-Boltzmann law's "funny prefactor" of 40.802624638, which exactly equals 2π⁵/15 and emerges from Planck's theory[^1]. The researchers created a framework for evaluating these potential connections by analyzing expressions built from common mathematical constants (π, e, γ, φ) using basic operations like addition, multiplication and exponents[^1].

[^1]: Surprises in numerical expressions of physical constants