Mersenne primes and the sequence $w_0=2,w_{n+1}=2w_n^2-1$

Question

Given the sequence $w_0=2, w_{n+1}=2w_n^2-1.$

It appears, from purely numeric examples, that we have this result:

When $q$ is an odd prime number, then $M_q=2^q-1$ is prime if and only if $M_q\mid w_{q-2}.$

I can actually prove if $M_q\mid w_{q-2}$ then $M_q$ is prime. [See below.] So the tricky part is if $M_q$ is prime, is $M_q\mid w_{q-2}?$

Can anybody find a proof or disproof of this?

I've tested this for the Mersenne primes $M_q$ with $q<10000.$

Some properties of $w_n$ are:

$w_n = \frac{1}{2}\left((2+\sqrt 3)^{2^n}+(2-\sqrt 3)^{2^n}\right)$
$w_n = T_{2^n}(2),$ where $T_k$ are the Chebyshev polynomials of the first kind.
The values of $w_n$ are pair-wise relatively prime.

This came about initially from this question. In my answer there, I show that if we have an odd prime factor $p\mid w_{n}$ then:

A. If $p\equiv 1\pmod{12}$ then $2^{n+2}\mid p-1.$
B. It is not possible that $p\equiv -1\pmod{12}.$
C. Otherwise $2^{n+2}\mid p^2-1.$

We can show that if $M_q\mid w_{q-2}$ then $M_q$ is prime using $(A)$ and $(C)$.

If $p\mid M_q$ and $M_q\mid w_{q-2}$ then by (C) and (A), you have that $2^q\mid p^2-1.$ This means that $2^{q-1}\mid p+1$ or $2^{q-1}\mid p-1.$ In either case, $p\geq 2^{q-1}-1=\frac{M_q-1}{2}.$ But this isn't possible unless $p=M_q$ since otherwise, $3p>M_q$ and $M_q$ is odd.

Also compare this to $2x^2+4x+1$ which doubles and adds 1 to the exponent if $x$ is a Mersenne. Hence iterating it starting with $x=1$ we get the Double Mersennes. — , Aug 28 '19 at 20:00
tempted to write long answer and show many other forms.https://math.stackexchange.com/questions/3143684/modular-arithmetic-rules-of-iteration-of-a-polynomial-function-are — , Aug 29 '19 at 11:47

score 2 · Accepted Answer · answered Aug 28 '19 at 19:20

2

I believe you have re-discovered the Lucas-Lehmer test. For a proof, see this Wikipedia article.

answered Aug 28 '19 at 19:20

J. W. Tanner

60,406

score 1 · Answer 2 · answered Aug 29 '19 at 12:32

This is what I call the reduced Lucas-Lehmer test. The original can also use start values in A018844 . Noting that all terms after the first in this reduced form of the LL test are odd, we can also reduce it to the $k$ values such that the sequence is $2k+1$ (as well as $3l+1$), and check if they are $M_{n-1}\bmod M_n$. This also goes to $32j+1$ etc (building the coefficient in powers of 2) as the terms increase in size. The $k$ values are linked by $k\to 4k^2+4k$ . We also have that any divisor of this reduced sequence, goes from 0 to -1 to 1 under mod which means we can use it to prove infinitely many normal primes. If we could force a proof of a new Mersenne as a prime factor, we'd prove infinitely many Mersenne primes (a currently unsolved problem). So there are quite a few possibly interesting implications from this sequence.

As to the proof, it follows from the fact that 2 is not a divisor of any Mersenne number, so it won't affect if it divides the sequence.

Mersenne primes and the sequence $w_0=2,w_{n+1}=2w_n^2-1$

2 Answers2