Characterizing a sequence of primes

Question

This is an attempt to finish up Characterizing the primes which don't divide any Pell-Lucas number(s)

For primes $p \equiv 3 \pmod 4,$ there is always some solution to $x^2 - 2 y^2 = \pm 1$ with $x \equiv 0 \pmod p.$ For primes $p \equiv 5 \pmod 8,$ never. For primes $p = 4 x^2 + 4 x y + 9 y^2 $ or $p = 4 x^2 + 4 x y + 33 y^2, $ always; these are $1 \pmod 8.$ Put another way, these predictable $1 \pmod 8$ primes are $p = u^2 + 8 v^2$ with $v$ required odd, then $p = u^2 + 32 v^2$ with $v$ required odd.

What I am left with is a separation of the primes $p = x^2 + 128 y^2$ into two subsets, one about a third of the primes and the other two thirds. The smaller set, up to 10,000, is

$$ 353, 1153, 1201, 1217, 1601, 2113, 2273, 3137, 4481, 5297, 5569, 6689, 7393, 7793, 8081, 8609, 9521, 9649, $$ The sequence of Pell-Lucas numbers is $P_1 = 1, \; P_2 = 3, \; P_3 = 7, \; P_4 = 17, \; P_{n+2} = 2 P_{n+1} + P_n. $ Note $P_{11} = 8119 = 23 \cdot 353$ and $P_{12} = 19601 = 17 \cdot 1153.$ The indexes for which these primes first occur as factors are

353 position   11
1153 position   12
1201 position   75
1217 position   76
1601 position   100
2113 position   44
2273 position   142
3137 position   196
4481 position   140
5297 position   331
5569 position   116
6689 position   418
7393 position   231
7793 position   487
8081 position   505
8609 position   269
9521 position   595
9649 position   603

The larger list of primes is $$ 137, 521, 569, 593, 857, 953, 1321, 1777, 1993, 2129, 2153, 2377, 2521, 2729, 2777, 2833, 3001, 3209, 3361, 3761, 3929, 4073, 4177, 4289, 4649, 4657, 4721, 4729, 4889, 4969, 5233, 5273, 5449, 5641, 5801, 6353, 6449, 6481, 7001, 7417, 7673, 8089, 8273, 8297, 8329, 8377, 9161, 9281, 9433, 9929, $$ These primes never divide any $u$ with $u^2 - 2 v^2 = \pm 1.$

Up to 1,000,000, there are 4891 primes $p = x^2 + 128 y^2,$ the smaller list has 1666 members, the larger 3225; then $3225/1666 \approx 1.936.$ Up to 10,000,000, there are 41,453 primes $p = x^2 + 128 y^2,$ the smaller list has 13,837 members, the larger 27,616; and $27616/13837 \approx 1.9958.$ Currently running 100,000,000, probably another six hours. Total was about nine hours; up to 100,000,000 there are 359,930 primes $p = x^2 + 128 y^2,$ the smaller list has 119,930 primes, the larger list 240,000, and $240000/119930 \approx 2.001167.$

Anyway, the kind of thing I now how to find is discriminants with, say, the principal genus having the number of classes divisible by 3, in which case the smaller list would be represented by the principal form, as in Cox's book. No luck here.

In comparison, the other side of the coin in Cox's book is a polynomial $f(x)$ for which given primes have a prescribed pattern of roots when considering $f(x) \pmod p.$ The favorable ratio of primes, so close to 2:1, then comes under the heading of Cebotarev Density.

So, there is the overall question, can anyone characterize these primes?

Answer 1

Found an acceptable answer, 1996 article by Pieter Moree, On the prime density of Lucas sequences, Journal de Theorie des Nombres de Bordeaux, volume 8 (1996) pages 449-459.

We find excerpts

ABSTRACT. The density of primes dividing at least one term of the Lucas sequence ... is determined.

the sequence ...$\{2,2,6,14,34, \cdots \},$ the so-called Pell sequence, plays an important role.

...This was first proved by Lagarias. Taking $P=2$ it is seen that the prime density of the Pell sequence is $17/24.$

So, for what I got, density $1/2$ for all primes $p \equiv 3 \pmod 4.$

Next, for positive binary quadratic forms we have $h(-512) = 8.$ Form Theorem 9.12 on page 188, with example $\Delta = -56$ on page 190, we add

$$ \langle 4,4,33 \rangle : \frac{1}{16}, $$ $$ \langle 9,\pm 8,16 \rangle : \frac{1}{8}, $$ $$ \frac{1}{3} \cdot \langle 1,0,128 \rangle : \frac{1}{3} \cdot \frac{1}{16} = \frac{1}{48}. $$ Then $$ \frac{1}{16} + \frac{1}{8} + \frac{1}{48} = \frac{10}{48} = \frac{5}{24}. $$ Finally $$ \frac{1}{2} + \frac{5}{24} = \frac{17}{24}, $$ which is what Moree requires. So, I think i got it. Have not proved everything, just tested on computer up to 1,000,000. Deterministic up to that point, the check comes under the heading of Pisano periods and is simply not difficult.

I still think that the subset of primes $p=u^2 + 128 v^2$ that work (go into the 17/24) can be described as the primes $1 \pmod 8$ for which some polynomial $f(x)$ of unpredictable degree $n,$ factors into $n$ linear factors, but I am the last person to actually find such a polynomial.