Constructing Carmichael Numbers with Certain Factors

Question

Construct a Carmichael number $p$ $=$ $1$ $\pmod n$ where all (prime) divisors $q$ $=$ $1$ $\pmod n$. For example how would one Construct a Carmichael number $p$ $=$ $1$ $\pmod {11}$ where all divisors $q$ $=$ $1$ $\pmod {11}$? Can we prove the existence of infinitely many of the specific Carmichael Numbers? Thanks for help.

Answer 1

I might be able to help with the first half. I see from http://mathworld.wolfram.com/CarmichaelNumber.html that numbers of the form $(6k+1)(12k+1)(18k+1)$ are Carmichael numbers if each of the three factors is prime. To attempt to find a Carmichael number meeting your requirements, calculate $6ni+1,12ni+1,$ and $18ni+1$ for various integer values of $i$. In your example of $n=11$, the above link says $k=55 (i=5)$ produces a Carmichael number.

Looking at http://oeis.org/A046025 yields more values: $k=121,506$ and $825$ for starters. A computer program could easily find more.