Counting Fibonacci primes: what a finite computation can and cannot add
A Fibonacci prime is a Fibonacci number that is also prime. The Fibonacci numbers are the familiar sequence 1, 1, 2, 3, 5, 8, 13, and so on, where each term is the sum of the two before it. Most of them are not prime. As of June 2025 we know of 57 positions in the sequence where the Fibonacci number is prime or almost certainly prime. They are spread very far apart. The largest one that has been fully proven prime sits at position 201107. The largest one we believe is prime sits at position 11964299, a number with about 2.5 million digits, checked by a fast test that is right in practice but does not constitute a proof. Whether the list goes on forever is an open question. No one knows a rule that produces these positions.
This study takes a different stance. It treats the 57 positions as a small dataset and asks a plain question. Is there any hidden pattern in them, and if there is, where does it live? The mathematics needed to answer this is already in the literature. What the study adds is method. It puts an honest error bar on a prediction, it runs a battery of tests to tell a real pattern apart from a coincidence, and it ships code so anyone can rerun everything. The scripts are at github.com/krimler/k-bonacci.
How many should there be
There is a well-known way to guess how many Fibonacci primes lie below some cutoff. The Fibonacci numbers grow at a steady multiplicative rate, so a Fibonacci number deep in the sequence has a predictable size. A number of that size is prime with a chance that shrinks as the number grows, in the usual way primes thin out. A classical divisibility fact narrows the search further. A Fibonacci number can only be prime when its position is itself prime. Combining these gives a clean prediction. The number of Fibonacci primes up to a cutoff grows in step with the logarithm of that cutoff, times a fixed constant.
This is the Grantham and Granville heuristic (Grantham and Granville 2024). Run it out to the current search depth and it predicts about 60 positions. We observe 57. The agreement is close.
There is a wrinkle worth airing. A quick fit of the constant comes out about 19% higher than the value the theory predicts. The study tracks the gap down. Fitting the constant against the Fibonacci data alone lands within 3% of the theoretical value. Pooling several related sequences gives a range that comfortably contains it. The apparent excess turns out to be a mix of a bookkeeping choice in the formula and ordinary small-sample noise. With only 57 points, the data cannot even tell the theoretical constant apart from a nearby look-alike value. Settling that would take far more data than the search has produced.
Predicting the next one
If the positions are spread out with no memory of each other, you can still say something useful about the next one. Look at the gaps between consecutive positions on a logarithmic scale. Those gaps behave like the waiting times in a process that has no memory, the same statistics that govern how long you wait for the next random arrival. Standard tests back this up.
That observation supports a prediction with an error bar, which the earlier work never attempted. From the data available up to a point, the method gives a 90% window for where the next position should fall. The position that was actually found next landed inside the window. Replaying this one step at a time across the whole list, the true position fell inside its predicted window about 87% of the time. For a method with a single tunable number, landing near the promised 90% is about as well as one can ask.
Where structure does live
The positions of the primes carry no usable structure. The way primes divide these sequences is a different story, and here the picture is exact.
For the Fibonacci numbers, ask which is the first position a given prime divides. A classical result of Lagarias says that for two thirds of primes that first position is even (Lagarias 1985). The study confirms this to high precision across many primes. The finer details of this divisibility follow from a known proof built on a symmetry argument from algebra (Cubre and Rouse 2014).
The study then moves to relatives of the Fibonacci numbers built from longer recurrences, where each term sums the previous three, four, or more. These no longer have the clean divisibility property. Their behaviour is instead controlled by a symmetry group attached to the recurrence, and a classical theorem turns that symmetry into exact proportions for how the sequence repeats when reduced by a prime. The study confirms these proportions numerically for the three-term and four-term cases.
For the three-term case there is one quantity the study can only estimate. It measures it at about 0.40 and is careful to flag that the exact value is not known. The study also corrects a claim in an earlier draft that said these finer densities were computable for any length of recurrence. For the general case they are open, with a partial result under a standard unproven hypothesis as the best that is currently known (Järviniemi 2022).
What is known and what is new
The point of the study is the line it draws between the two. The count estimate, the matching constant, the two-thirds divisibility result, the symmetry-driven proportions, and the repetition structure of the longer sequences are established mathematics with citations. The new contributions are smaller and clearly marked. There is a prediction window with a measured success rate. There is a reusable battery of tests for telling structure apart from noise. There is a fully cited reproduction that anyone can run.
Reproducibility
Every result comes from self-contained Python scripts using standard scientific libraries. The input data are the published index lists for the Fibonacci, Lucas, and related sequences. The code is at github.com/krimler/k-bonacci.
Read the paper
The full paper, with the precise statements, the constants, and the proofs it draws on, is here: Fibonacci and k-bonacci prime statistics (PDF and details).