In my original post I stated that the upper bound was it is in fact , and here's the proof you requested jamix:

Theorem: Let be a Carmichael number with r,s, and t prime. Without loss of generality, we assume that . The following then hold:

(i)

(ii)

(iii)

Proof: Let be a Carmichael number of three factors. We know that for i = {1,2,3}. We can then see that:

and

These imply that:

for any

and

for any

Since we said that we can then say that , in particular:

because

contradicts the primality of s.

Same for q and 1(t - 1).

So, we get:

Putting the equation for t into the equation for s, we get:

This implies that:

This gives you:

This result tells you that because the opposite yields and

which is not in .

We are then going to say that because and

This will then give us:

It is common in most proofs to now review by using positive real numbers x, y , and z where , we then have:

Since we assumed that we have hence . We can then say that:

This gives us:

Assertion 1.

From and we get:

Assertion 2.

And finally, we get:

Assertion 3.

As desired. Pardon my horrible symbols, that's how I write it down... Sorry about that.