Eccoci.

When we calculate $8^n$ modulo $11$ for $n=0$ to $9$ we get: $1, 8, 9, 6, 4, 10, 3, 2, 5, 7$.
As we see all possible values from $1$ to $10$ occur. So $8$ is a primitive root of $11$.
But there is more:
If we take a closer look we see:
$1+8=9$
$8+9=17 \equiv 6 \bmod 11$
$9+6=15 \equiv 4 \bmod 11$
$6+4=10$
$4+10=14 \equiv 3 \bmod 11$
$10+3=13 \equiv 2 \bmod 11$
$3+2=5$
$2+5=7$
$5+7=12 \equiv 1 \bmod 11$.

So the powers of $8 \bmod 11$ are cyclic with period $10$, and $8^n + 8^{n+1} \equiv 8^{n+2} \pmod{11}$.
$8$ is called a Fibonacci primitive root of $11$.
Not every prime has a Fibonacci primitive root.
There are $323$ primes less than $10000$ with one or more Fibonacci primitive roots and the sum of these primes is $1480491$.
Find the sum of the primes less than $100\,000\,000$ with at least one Fibonacci primitive root.