Presumably that's the density of S qua subset of primes, of which 172 are < 1024 against 18 in my list; and 172/8 = 21.5 , in tolerable agreement with experiment. And now a well-aimed boot up the backside from an eminent colleague discloses that S comprises exactly those primes represented by binary quadratic form x^2 + 14*y^2 , see OEIS A033211. I don't properly understand the connection at this stage, but it presumably involves biquadratic residues (ugh!). WFL On 5/20/18, Victor Miller <victorsmiller@gmail.com> wrote:
S has asymptotic density 1/8 by the Chebotarev density theorem since the Galois group of the polynomial has order 8.
On Sat, May 19, 2018 at 20:13 Fred Lunnon <fred.lunnon@gmail.com> wrote:
The quartic polynomial z^4 - 2*z^3 + z^2 - 2*z + 1 factorises completely into linear factors over finite field |F_p for p in set S = { 23, 127, 137, 151, 233, 239, 281, 359, 431, 449, 487, 673, 743, 751, 911, 953, 967, 977, ... } (all members with p < 1024).
Can anything constructive be said about S ? For instance,
Is S infinite?
Does S contain subsets of form { p | p prime & p == a (mod b) } ?
What (bounds on) asymptotic density has S , if any?
WFL
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