## Fibonacci primitive roots

### Problem 437

Published on Saturday, 21st September 2013, 10:00 pm; Solved by 343
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≡6 mod 11

9+6=15≡4 mod 11

6+4=10

4+10=14≡3 mod 11

10+3=13≡2 mod 11

3+2=5

2+5=7

5+7=12≡1 mod 11.

^{n}+ 8

^{n+1}≡ 8

^{n+2}(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 100,000,000 with at least one Fibonacci primitive root.