New project : Define $$f(n):=\prod_{j=1}^n F_j$$ where $F_j$ denotes the $j$-th Fibonacci-number. $f(n)-1$ is prime for $n=4,5,6,7,8,14,15$ and $f(n)+1$ is prime for $n=1,2,3,4,5,6,7,8,22,28$. Are there more primes of the form $f(n)\pm1$ ?

@MartinHopf

@TeresaLisbon

@Peter Nice! This one seems to be interesting, till where have you checked both hypotheses?

The "-" case and the "+" - case. Unfortunately, we can only use PARI/GP for this calculation , unless someone knows a trick.

@Peter I'm sorry I wasn't clear, I meant : till what $n$ did you check the primality of $f(n)-1$ and $f(n)+1$?

The current program is at $n=360$, but with pre-sieving I will be able to get a better speed. But I like doublechecks...

Ok, $n=360$ is very large, we should check if there's like a list of primes common among a certain bunch of multiples . For example , is there a predictable sequence of $n$ for which $f(n)+1$ or $f(n)-1$ is a multiple of $3/5/7$ etc.

We can then start ruling out all such $n$. Hopefully there will be some modulo pattern to help us.

Rather the opposite holds. Every prime factor of a fibonacci-number upto $F_n$ CANNOT be a prime factor of either $f(n)-1$ or $f(n)+1$. So, we have many candidates.

But the sequence $f(n)$ grows of course very quickly. Hence it can take a while to get the next PRP.

Oh, I see. Oh dear, this is a problem.

The lack of predictability means that we'll have to check every value of n, unfortunately.

To be more precise : Presieving will not remove many candidates, but some gain of speed might be possible. Maybe , PFGW allows to eveluate products , factordb seems not to support it.

Maybe, someone in the mersenne forum knows a syntax for PFGW.

I do not think that there is a closed form expression readable by factordb or PFGW, right ?

No, I don't think so.

It's quite an abstract expression that we are dealing with right now.

If you are patient enough to input long strings as I(1)*I(2)*...*I(500) , you can in principle use factordb.

Yes, you can create a program to generate such strings for you, which you can then copy-paste into factordb.

and then delete one factor by another beginning with the longest. Would you be patient enough for this job ?

Sure, why not? Let me know what exactly to print out, I can get it done and let you know how it's done.

I mean to say : I'll create the program and find you an online IDE where you can run it get the output and put it into factordb. Python will do the string-generation very easily.

To save time and to avoid typos, I can try to write a program creating the desired string.

If you can do it, that would be great as well. Let me know what I need to do for assistance.

This seems to do the job :

k=200;a="";for(s=1,k,a=concat(a,"I(");a=concat(a,Str(s));a=concat(a,")");if(s<k,a=concat(a,"*")));print(a)

Yes, this looks like it does the job as well! In fact ,this is really nice and self-contained.

start with $n = 1\ 000$

I am at $n=409$

We will see whether factordb can handle such long string at all.

Ok

By the way, I need to leave, so you can leave the results here and I will definitely take a look at them tomorrow. Meanwhile, my thread will be free tomorrow so I'll be able to work with yafu and LLR if required. Thanks.