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:
$1+8=9$
$8+9=17\equiv 6mod11$
$9+6=15\equiv 4mod11$
$6+4=10$
$4+10=14\equiv 3mod11$
$10+3=13\equiv 2mod11$
$3+2=5$
$2+5=7$
$5+7=12\equiv 1mod11$.

So the powers of $8mod11$ are cyclic with period $10$, and $8^n+8^{n+1}\equiv 8^{n+2}(\mathrm{mod}11)$.
$8$ 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 $100000000$ with at least one Fibonacci primitive root.