# Does some Lucas sequence contain infinitely many primes?

Does some nontrivial Lucas sequence contain infinitely many primes?

The Mersenne numbers $$M_n=2^n-1:n$$ not necessarily prime are a Lucas sequence with recurrence relation $$x_{n+1}=2x_n+1$$.

It's an open problem how many Mersenne numbers are prime and we know neither whether $$0\%$$ or $$100\%$$ are prime (asymptotically speaking).

There are also similar sequences of repunits base $$n$$ with some nice maths surrounding them.

There are Lucas sequences having primes up to some point and then no more primes, such as the sequence with the relation $$x_{n+1}=4x_n+1$$ given by $$1,5,21,85,341,\ldots$$ for which it can be shown that there are no more primes beyond $$5$$.

We can also find sequences having no primes at all such as the sequence with the same relation but starting at $$8$$, given by $$8,33,133,533,\ldots$$ - and in fact it is true for any sequence for which $$3x_0+1$$ is a square that it has no primes - so we can say there are infinitely many Lucas sequences having no primes.

The obvious case to ask is whether infinitely many of the Fibonacci numbers are prime - and this is another open problem.

Is it known, or is it possible to show, that there is some (nontrivial) Lucas sequence (identifiable or otherwise), having infinitely many primes, or that there is none?

• @vadim123 I could've sworn I put "nontrivial" in the question! I must've edited it out - I've put it back in, sorry. – user334732 Nov 3 '18 at 19:21
• Also, Lucas sequences are normally (a) second-order; and (b) nonhomogeneous. Neither of the two examples given satisfy those criteria. – vadim123 Nov 3 '18 at 19:22
• @vadim123 I can't find any reference of what you mean there. By 2nd order does this mean expressing $U_{n+2}$ in terms of $U_n$ and $U_{n-1}$ as done here: math.stackexchange.com/questions/2705983 rather than expressing $U_{n+1}$? – user334732 Nov 4 '18 at 12:07
• – vadim123 Nov 4 '18 at 22:15
• @vadim123 thanks, plenty to digest there, I'll work through it. I'd come across characteristic polynomials before and wanted to understand better too. It's a long shot but do you know of some obvious link between these characteristic polynomials and the Cantor pairing function? My particular interest is whether a certain form of power series may be the limit of polynomial pairing (tuple) functions as the degree approaches infinity. – user334732 Nov 4 '18 at 22:32