DougMerritt (log😅 = 💧log😄) (@dougmerritt.mathstodon.xyz.ap.brid.gy)

More on Carmichael This morning I took an old blog post and turned it into an X thread. I think the thread is easier to read. More expository and less rigorous. The post and thread look at generalizations of the fact that every integer and its fifth power end in the same digit. The conclusion is that _n_ and _n_ _k_ end in the same digit base _b_ if _b_ is square-free and _k_ = φ(_b_) + 1. So, for example, 10 is square-free (i.e. not divisible by a non-trivial square) and φ(10) + 1 = 5. Benjamin Clark replied suggesting looking at λ(_b_) + 1 in addition to φ(_n_) + 1. ## Euler and Carmichael To back up a bit, φ(_n_) is Euler’s totient function, the number of positive integers less than and relatively prime to _n_. Robert Carmichael’s totient function λ(_n_) is a closely related function, one that replaced Euler’s function in implementations of RSA encryption—more specifically in RSA _de_ cryption—something I wrote about earlier this week. Euler’s generalization of Fermat’s little theorem says if _a_ is relatively prime to _m_ , then _a_ φ(_n_) = 1 (mod _n_). Carmichael’s totient λ(_n_) is the smallest number such that _a_ λ(_n_) = 1 (mod _n_) when _a_ is relatively prime to _n_. Sometimes φ(_n_) = λ(_n_), namely for _n_ = 1, 2, 4, and every odd prime power. Otherwise λ(_n_) is a proper divisor of φ(_n_). ## Carmichael’s conjecture Carmichael conjectured that for every _n_ there is at least one other integer _m_ ≠ _n_ such that _φ_(_m_) = _φ_(_n_). He proved that this is true at least for _n_ less than 1037. The conjecture is now known to be true for _n_ less than 101010. https://www.johndcook.com/blog/2025/10/09/more-on-carmichael/