A047650 Primes for which golden mean tau is a quadratic residue.

29, 89, 101, 181, 229, 349, 401, 461, 509, 521, 541, 709, 761, 769, 809, 941, 1009, 1021, 1049, 1061, 1109, 1229, 1249, 1289, 1361, 1409, 1549, 1601, 1621, 1669, 1709, 1721, 1741, 1789, 1861, 2029, 2069, 2081, 2089, 2389, 2441, 2621, 2729, 2801, 2861

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Primes of the form x^2 + 20*y^2. - T. D. Noe, May 08 2005

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

From A.H.M. Smeets, Nov 16 2023: (Start)

Mean gap size between two consecutive terms at p: ~ 8*log(p).

In x^2 + 20y^2: x == 1 (mod 2) and x !== 5 (mod 10). Otherwise not prime. (End)

Links

• A.H.M. Smeets, Table of n, a(n) for n = 0..20000 (terms 0..1700 from Vincenzo Librandi)
• Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
• E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138.
• E. Lehmer, Correction, Fib. Quart., 4 (1966), 135-138.
• E. Lehmer, On the quadratic character of the Fibonacci root (annotated scanned copy)

Crossrefs

Cf. A047651, A001583.

Cf. A005968.

Programs

Magma

k:=20; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Vincenzo Librandi, Sep 05 2016

Mathematica

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 *)

Formula

From A.H.M. Smeets, Nov 16 2023: (Start)

Equals {prime(n): A296240(n) in {2^k: k > 0}} = {A308787} union {A308789} union {A308793} union ... .

a(n) ~ A000040(8*n). (End)

Extensions

More terms from James Sellers, Jan 25 2000