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 nk 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 decryption—something I wrote about earlier this week.
Euler’s generalization of Fermat’s little theorem says if a is relatively prime to m, then
aφ(n) = 1 (mod n).
Carmichael’s totient λ(n) is the smallest number such that
aλ(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.