Lehmer's totient problem is an well-known open problem which states that the divisibility :
$$\phi(n) \mid n-1$$ holds only for primes .
This motivated me to ask the following :
For which integers $k$ we can find a composite $n$ such that $$\phi(n)+k \mid n-1+k$$
Obviously we can reduce the divisibility to $$\phi(n)+k \mid n-1-\phi(n)$$ For example trying with some composites :
- $n=4$ gives $2+k \mid 1$ which leads to $k=-1$ and $k=-3$
- $n=6$ gives $2+k \mid 3$ which leads to $k=1,-1,-3,-5$
- $n=9$ gives $6+k \mid 2$ which leads to $k=-4,-5,-7,-8$
There seems to be a lot of values that work .I am pretty confident that all the values except $0$ will appear at some point :
Conjecture All $k$'s except $0$ have this property .
This seems true but I'm completely stuck on how to prove such a thing .
I would love to see a proof of this fact (if it's even true ) .
Thanks a lot for your time to help me with this .I really appreciate all your help.