Short – but not irrelevant – exercise February 7, 2007Posted by dorigo in mathematics, physics, science.
You are given three different positive real numbers A<B<C (C is the largest). Each of them has a uncertainty equal to a part in ten thousand or smaller. How large does the integer number N=1/Q have to be in order for the following formula to hold:
(A+B+C)/[(A^Q + B^Q + C^Q)^N] – 1 < 10^(-5) ,
as a function of the ratio C/B?
The question has some relevance to the strange coincidence found by Koide on charged lepton masses. I was startled by the astounding coincidence between that ratio (save a probably well-motivated factor of 2/3 at the denominator) and unity (the most up-to-date number is 1. +- 0.000026), but then I started thinking at what really is the probability of something like that occurring by chance.
Assessing the real “significance” of such a coincidence -that is, assessing how likely it is that such a simple formula can describe the ratios of lepton masses, numbers that the Standard Model does not explain and which appear to be falling out of nowhere, is important, but probably unfeasible. In fact, one would then also need to estimate how likely it is that a formula describing “by chance” these ratios can be obtained by using eigenvectors of a quite simple 3×3 matrix.
Despite the trouble with assessing the probability of the coincidence stated above, it remains in my opinion a quite interesting open question.