A099011 Pell pseudoprimes: odd composite numbers n such that P(n)-Kronecker(2,n) is divisible by n.

169, 385, 741, 961, 1121, 2001, 3827, 4879, 5719, 6215, 6265, 6441, 6479, 6601, 7055, 7801, 8119, 9799, 10945, 11395, 13067, 13079, 13601, 15841, 18241, 19097, 20833, 20951, 24727, 27839, 27971, 29183, 29953, 31417, 31535, 34561, 35459, 37345

Offset: 1

Author

Jack Brennen, Nov 13 2004

Keywords

• nonn

Comments

Here P(n) are the Pell numbers (A000129), defined by P(0)=0, P(1)=1, P(x) = 2*P(x-1) + P(x-2) and Kronecker(2,n) is equal to 1 if n is congruent to +-1 mod 8 and equal to -1 if n is congruent to +-3 mod 8.

Examples

			169 is a Pell pseudoprime because P(169)-Kronecker(2,169) is divisible by 169.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site, terms 1..200 from Ralf Stephan)
• Antonio J. Di Scala, Nadir Murru, and Carlo Sanna, Lucas pseudoprimes and the Pell conic, arXiv:2001.00353 [math.NT], 2020.
• Dana Jacobsen, Pseudoprime Statistics, Tables, and Data (includes terms to 5e12)

Crossrefs

Cf. A000129.

Programs

Mathematica

pell[0] = 0; pell[1] = 1; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellpspQ[n_] := OddQ[n] && CompositeQ[n] && Divisible[pell[n] - JacobiSymbol[2, n], n]; Select[Range[40000], pellpspQ] (* Amiram Eldar, Nov 22 2019 *)

PARI

genit(nterms=100)={my(arr=List(),q,z);forcomposite(n=9,+oo,if(n%2==0,next);q=([2,1;1,0]^n)[2,1];z=(q-kronecker(2,n))%n;if(z==0,listput(arr,n));if(#arr>=nterms,break));arr} \\ Bill McEachen, Jun 24 2023 (incorporates A000129 code of Greathouse)

Perl

use Math::Prime::Util qw/:all/;
my($U,$V);
foroddcomposites {
  ($U,$V) = lucas_sequence($, 2, -1, $);
  $U = ($U - kronecker(2,$)) % $;
  print "$_\n" if $U == 0;
} 1e11; # Dana Jacobsen, Sep 13 2014