If, like me, you consider yourself a “serious” ( ™ ) arithmetician, you may also have started spurning at an early age the type of numerical coincidences, sometimes found in dictionaries of remarkable numbers, that state that, for instance, is the only integral solution to the equation
However, maybe I shouldn’t be so dismissive. Today, after a discussion with P-O Dehaye, which was certainly serious enough since the Ramanujan function featured prominently, the question came to understand the solutions in of the system
(everything modulo ).
This sounds rather innocuous, but something I find rather amusing happens: you can check for yourself that the solutions are either (i) diagonal (all are equal) or (ii) a permutation of (i.e., no two coincide). Why is that? Well, first both (i) and (ii) do describe solutions. Clearly (( ™ ) again) there are solutions in total; there are diagonal ones, and permutations; since
there can be no other solution.
Is there a more direct proof?