Math Outside the Bubble: Has Carmichael’s TotientConjecture Been Proven?

摘要

Carmichael’s totient conjecture lacks the name recognitionof such media darlings as the twin prime conjecture orthe Riemann hypothesis, but in 2018 this open questionin number theory blipped briefly into the public consciousness.It cropped up in, of all places, a DriveTime?commercial.Besides bearing a name beginning with the syllablecar, Carmichael’s totient conjecture (CTC) has nothing todo with DriveTime’s stock-in-trade, which is the sale andfinancing of used automobiles. CTC concerns the multiplicityof values of Euler’s totient function. Euler’s totientfunction (n) returns for a positive integer n the numberof positive integers at most n that are relatively prime to n(where 1 is counted as being relatively prime to all numbers).CTC posits that for every n∈ + there is at least onem∈ +, m≠n such that (m)= (n).