|
%I #66 Dec 21 2023 15:08:56
%S 29,89,101,181,229,349,401,461,509,521,541,709,761,769,809,941,1009,
%T 1021,1049,1061,1109,1229,1249,1289,1361,1409,1549,1601,1621,1669,
%U 1709,1721,1741,1789,1861,2029,2069,2081,2089,2389,2441,2621,2729,2801,2861
%N Primes for which golden mean tau is a quadratic residue.
%C Primes of the form x^2 + 20*y^2. - _T. D. Noe_, May 08 2005
%C Also primes p that divide the sum of cubes of the first (p-1)/2 Fibonacci numbers A005968((p-1)/2). - _Alexander Adamchuk_, Aug 07 2006
%C From _A.H.M. Smeets_, Nov 16 2023: (Start)
%C Mean gap size between two consecutive terms at p: ~ 8*log(p).
%C In x^2 + 20y^2: x == 1 (mod 2) and x !== 5 (mod 10). Otherwise not prime. (End)
%H A.H.M. Smeets, <a href="/A047650/b047650.txt">Table of n, a(n) for n = 0..20000</a> (terms 0..1700 from Vincenzo Librandi)
%H Bob Bastasz, <a href="https://www.fq.math.ca/Papers1/58-5/bastasz.pdf">Lyndon words of a second-order recurrence</a>, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
%H E. Lehmer, <a href="http://www.fq.math.ca/Scanned/4-2/lehmer.pdf">On the quadratic character of the Fibonacci root</a>, Fib. Quart., 4 (1966), 135-138.
%H E. Lehmer, <a href="http://www.fq.math.ca/Scanned/4-4/correction.pdf">Correction</a>, Fib. Quart., 4 (1966), 135-138.
%H E. Lehmer, <a href="/A001583/a001583.pdf">On the quadratic character of the Fibonacci root</a> (annotated scanned copy)
%F From _A.H.M. Smeets_, Nov 16 2023: (Start)
%F Equals {prime(n): A296240(n) in {2^k: k > 0}} = {A308787} union {A308789} union {A308793} union ... .
%F a(n) ~ A000040(8*n). (End)
%t nn=20; pMax=3000; Union[Reap[Do[p=x^2 + nn*y^2; If[p<=pMax&&PrimeQ[p], Sow[p]], {x, Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2, 1]]] (* _Vincenzo Librandi_, Sep 05 2016 *)
%o (Magma) k:=20; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // _Vincenzo Librandi_, Sep 05 2016
%Y Cf. A047651, A001583.
%Y Cf. A005968.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Jan 25 2000
|
