Number of Fibonacci primes

A “Fibonacci prime” is nothing more than a number which is both prime and a Fibonacci number. The indices of the first 33 Fibonacci primes, and the “probable” next 13 (in the probable prime sense), are A001605 in the On-Line Encyclopedia of Integer Sequences. The sequence of such indices begins 3, 4, 5, 7, 11, 13, …; the corresponding Fibonacci numbers are 2, 3, 5, 13, 89, 233, … . As is pointed out in the note at the OEIS, since F_mn is divisible by F_n all further entries in that sequence must be prime. The converse is not true; the first exception is F₁₉ = 4181 = (37)(113).

Are Fibonacci primes rare? To answer this question we need to define “rare”; specifically, if you know a number is a Fibonacci number, does that make it more or less likely to be prime than it would otherwise be? The nth Fibonacci number is roughly φⁿ, so by the prime number theorem the “probability” that the n

Fibonacci number is prime is 1/(n log φ). Integrating, there ought to be about (log n)/(log φ) primes among the first n Fibonacci numbers; inverting, the xth Fibonacci prime should be somewhere around index φ^x.

We can plot the points (1, log(3)), (2, log(4)), (3, log(5)), (4, log(7)), …, (46, log(1636007)), where the y-coordinates are the indices of the Fibonacci primes, and see if they fall roughly on a straight line. The picture is below. If the heuristic above is correct, they should fall on a line of slope log φ, or about 0.48. They actually fall on a line of slope about 0.31. For example, there should be, by the heuristic in the previous paragraph, about (log 1636007)/(log φ) ≈ 30 Fibonacci primes with index less than 1636007; there are actually 46. There are too many Fibonacci primes.

More generally, do Fibonacci numbers have fewer factors than are typical among numbers of their size? Here are factorizations of Fibonacci numbers at mersennus.net. I don’t know the answer to this question.

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