User:D4nn0v/sandbox

The proofs show that the Carmichael theorem/function/formula yields numbers that are the orders of multiplicative groups modulo n but do not explicitly show that these are the smallest possible sizes of these groups. I believe with some small modifications, it may be possible to make the proofs complete.

Proof for twice the power of an odd prime
This proof is very similar to the one in the previous section.

From Fermat's little theorem, we have $$a^{p-1}=1+hp$$. Because $$a$$ is coprime to $$2p$$, it must be an odd number. Thus, we can also write $$a^{p-1}=1+2lp$$ for some integer $$l$$. For $$k = 1$$, $$l=$$ some integer:


 * $$a^{p^{k-1}(p-1)}=1+l2p^k$$


 * $$a^{p^k(p-1)}=(1+l2p^k)^p=\sum_{i = 0}^p \binom{p}{i} (l2p^k)^i=1+p \times l2p^{k+1}+\dots=1+l_02p^{k+1}$$

for some integer $$l_0$$.

By induction, we have $$a^{p^{k-1}(p-1)}\equiv 1\pmod{2p^k}$$.