Fibonacci primes vs Mersenne primes

Question

It seems that only 34 Fibonacci primes are known while 54 Mersenne primes are known, while Fibonacci numbers are sparser than Mersenne numbers. Compare https://en.wikipedia.org/wiki/Fibonacci_prime and https://en.wikipedia.org/wiki/Mersenne_prime Is there a heuristic argument to explain this discrepancy?

Answer 1

The Fibonacci numbers, as well as the Mersenne numbers, are both members of the family of Lucas sequences. Presently, there is no known Lucas sequence (or companion Lucas sequence) that has been shown to contain an infinite number of primes. Nevertheless, I was reading recently in the 2nd edition of Richard K. Guy's ``Unsolved Problems in Number Theory'' that it seems to be widely conjectured that many are of the mindset that many of what Guy refers to as Lucas-Lehmer sequences do contain infinitely many prime terms.

I would suspect that most mathematicians who have given it soemserious thought probably think as well that the specific sequences you mention contain an infinite number of primes, Actually, in the case of the Mersenne numbers, I am quite certain of it.

That being said, if it is true that both of these sequences contain an infinite number of primes, without being given any additional information, what we do know is that primes in both of these sequences will occur only at terms whose index is prime.

Now, more Mersenne primes that Fibonacci primes have been discovered to date probably because of the unified interest (i.e., GIMPS) in these numbers. GIMPS has made it so simply to look for them that even a child of elementary school age can participate in the search for them. Also, the monetary prizes that were attached in the past for the discovery of the first of a above a given magnitude, as well as the attractive bounties that are still are in place today for the discovery of Mersenne primes of a hundred million digits or more and one billion digits or more add further incentive to maintaining interest in these primes.

Such to my knowledge is not the case for Fibonacci primes.

The largest Fibonacci prime discovered to date has six digits in its index (104911); on the other hand, the most recently discovered Mersenne prime contains eight digits in its exponent (82,589,933). Again, this is surely attributable to the concerted effort being waged for the discovery of the latter type.

Finally, although the structure of Mersenne primes is and appealing because of its base (i.e., 2^{p} - 1) and the ease by which they are converted into a binary representation consisting of $p$ $1s$, the Fibonacci primes are obtained from Binet's formula, which involves the golden mean. Hence, when studying Fibonacci numbers, there are countless upon countless upon countless of interesting things to try to prove and draw conclusion from; whereas, with Mersenne numbers, the big thing on the minds of those interested in them seems often enough just their primality.