# What is the first 11 digit prime of Fibonacci?

I am having difficulty finding the first eleven-digit prime number of Fibonacci..

If anyone has an answer I would greatly appreciate it. I'm mostly asking this because it's one part of a greater puzzle I am trying to solve..

• There are no eleven-digit Fibonacci primes: oeis.org/A005478 There is, however, a 10-digit Fibonacci prime: $2971215073$ – Noble Mushtak Dec 28 '18 at 23:07
• @RobArthan That's not how upvotes work. If you feel that this post is a good post (i.e. that it includes things like showing genuine effort, either work or research, and some context so that you, for instance, don't have to guess whether it's a coding problem) you are free to upvote. But they do not cancel eachother out, and they're not meant to. – Arthur Dec 28 '18 at 23:16
• @RobArthan I can see that it's one vote up and one down, rather than none. And you get 5 reputation for an upvote and lose 2 for a downvote on a question post. So no, they do not outweigh one another at all. I agree that downvoters ought to explain their reasons. But undoing votes is a moderator privilege (at least I think they can do that), and not something we should go around doing all willy nilly. Upvote if you genuinely think the post deserves it (I don't think it does). Not to undo a downvote. – Arthur Dec 28 '18 at 23:31
• @RobArthan Also, sociopath is a very strong word. Please refrain from using the names of actual mental diagnoses on what's likely just laziness or apathy. Or possibly being a jerk. – Arthur Dec 29 '18 at 0:22
• Alright for those who were wondering. An employer posted a puzzle to find the elusive 11 digit Fibonacci Prime. in regards to a network security job. Thing is I know the tools I just dont know the Fibonacci. So I came here seeking out some help. – CodeMonkeyAlx Dec 29 '18 at 0:49

It is a standard result that if the $$n$$-th Fibonacci number is prime, then $$n$$ is prime, unless $$n=4$$. This makes the search much easier.

The Fibonacci number $$F_{47}$$ has 10 digits, and the Fibonacci number $$F_{59}$$ has 12 digits. (this is easy to check by direct computation, or using the approximation $$F_n \approx \phi^n / \sqrt{5}$$.) There is only one prime between $$47$$ and $$59$$, namely $$53$$.

So all we have to check is whether $$F_{53}$$ is prime. But with a computer, it is easy to find that $$953$$ is a factor.

We can conclude that there are no 11-digit Fibonacci primes.

Here are the Fibonacci primes:

Select[Table[Fibonacci[i], {i, 1, 200}], PrimeQ]


$$\{2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437,$$

$$2971215073, 99194853094755497, 1066340417491710595814572169,$$

$$19134702400093278081449423917 \}$$

• OP asked for an 11 digit prime number. The fact that your number ends in $5$ may suggest that you have made a mistake. – Mohammad Zuhair Khan Dec 28 '18 at 23:57
• @OscarLanzi Please check the edit history. – Mohammad Zuhair Khan Dec 29 '18 at 0:10