## On twin primes and Lucas pseudoprimes

The OEIS is fantastic of course.  Lots of contributions (from professionals and amateurs alike), many interesting conjectures, and thus nice opportunies to learn more.

Here’s one conjecture though which unfortunately doesn’t work out (feel free to edit the corresponding entry A113910 accordingly). I’ll try to reconstruct here how I think the conjecture was made, and how it breaks down.

Lucas numbers are given by the second-order linear reccurence: $L_0=2$, $L_1=1$ and $L_{n}=L_{n-1}+L_{n-2}$.  There’s an explicit formula for the n-th term involving the golden ratio, $L_n=\left ( \frac{1+\sqrt{5}}{2}\right )^n+\left ( \frac{1-\sqrt{5}}{2}\right )^n$.

An interesting property is that whenever $n=p$ is a prime, one can prove that $L_p\equiv 1 \pmod{p}$ (see the first few pages of Sun’s review on congruences of Lucas numbers).   But this happens for some composite integers too, which have hence been named Lucas pseudoprimes. Ok, so for those integers $n$ (primes and Lucas pseudoprimes) we have defined some $A_n$ such that $L_n=1+n\times A_n$.

Now, on the OEIS page about Lucas numbers there’s a remark by Ward that

$L_n$ is the number of points of period n in the golden mean shift. The number of orbits of length n in the golden mean shift is given by the n-th term of the sequence A006206.

Indeed, recall that the shift $\sigma_\varphi$ associated to the golden ratio $\varphi:=\frac{1+\sqrt{5}}{2}$  is $\sigma_\varphi : X\rightarrow X$, where $X$ is the space of sequences made of 0′s and 1′s such that no consecutive 1′s appear.    This is equivalent to a dynamics modulo 1, namely $x\rightarrow \varphi x \pmod{1}$, if we code by 0 the interval $[0;\frac{1}{\varphi}]$ and by 1 the interval $[\frac{1}{\varphi};1]$. (See the Scholarpedia article by Marcus and Williams for references).

Moreover, on the page about A006206 one finds the remark that $L_n=\sum_{d|n} d\times B_d$, where I’ve noted $B_n$ the terms of A006206.  So, implicitely, this means that whenever $n=p$ is prime we have $L_p=1+p\times B_p (*)$ .

Probably this last observation is what lead Dement to his conjecture linking twin primes and Lucas numbers as mentionned in sequence A113910 (I’m rephrasing it slightly):

for $n>4$, if $\frac{L_{n+1}-B_{n+2}}{B_{n+2}-B_n} (**)$ is an integer then it is a prime, and those primes are exactly the lesser of a twin pairs (i.e. a characterization of twins).

The conjecture is made by remarking that if $p$ and $p+2$ are twin primes, well they are both primes so satisfy $(*)$, and then using the Lucas reccurence relation one obtains that $(**)$ is an integer, and the conjecture asks whether the converse holds.

But there’s the problem. Just forgetting what $B_n$ is, it’s clear that $B_n=A_n$ when defined, so the conjecture asserts that Lucas pseudoprimes (for which $(**)$ is also an integer) are never at a distance equal to 2 of a prime (i.e are never creating a pair n, n+2 which would pass the $(**)$ integrality test).

Unfortunately 5777 is a Lucas pseudoprime, and 5779 is a prime.

One great thing I’ve been made aware of while looking at that stuff are the results of Lagarias which in particular say that the set of primes which divide some Lucas number has density 2/3  , and that more work by various authors has been done since.