The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A120947

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A120947

a(n) = smallest m such that n-th prime divides Pell(m).
(history; published version)

#28 by OEIS Server at Fri Aug 28 21:10:39 EDT 2015

LINKS

Alois P. Heinz and Robert Israel, <a href="/A120947/b120947_1.txt">Table of n, a(n) for n = 1..10000</a> (n = 1 .. 1000 from Alois P. Heinz)

#27 by N. J. A. Sloane at Fri Aug 28 21:10:39 EDT 2015

STATUS

reviewed

approved

Discussion

Fri Aug 28

21:10

OEIS Server: Installed new b-file as b120947.txt.  Old b-file is now b120947_1.txt.

#26 by G. C. Greubel at Fri Aug 28 16:14:32 EDT 2015

STATUS

proposed

reviewed

#25 by Robert Israel at Fri Aug 28 16:07:16 EDT 2015

STATUS

editing

proposed

#24 by Robert Israel at Fri Aug 28 16:07:12 EDT 2015

LINKS

Alois P. Heinz and Robert Israel, <a href="/A120947/b120947_1.txt">Table of n, a(n) for n = 1..10000</a> (n = 1 .. 1000 from Alois P. Heinz)

#23 by Robert Israel at Fri Aug 28 16:06:33 EDT 2015

LINKS

Alois P. Heinz, and Robert Israel, <a href="/A120947/b120947_1.txt">Table of n, a(n) for n = 1..10000</a> (n = 1 .. 1000</a> from Alois Heinz)

#22 by Robert Israel at Fri Aug 28 16:03:33 EDT 2015

COMMENTS

For n > 1, a(n) is the multiplicative order of -3-2*sqrt(2), in GF(prime(n)) if 2 is a quadratic residue (mod prime(n)) or GF(prime(n)^2) otherwise. Thus a(n) divides prime(n)-1 if prime(n) == 1 or 7 (mod 8), i.e. n is in A024704, and a(n) divides prime(n)+1 if prime(n) == 3 or 5 (mod 8), i.e. n is 2 or is in A024705. - Robert Israel, Aug 28 2015

CROSSREFS

Cf. A000129 (Pell numbers), A001602 (equivalent sequence with Fibonacci numbers), A239111, A024704, A024705.

#21 by Robert Israel at Fri Aug 28 15:55:29 EDT 2015

COMMENTS

For n > 1, a(n) is the multiplicative order of -3-2*sqrt(2), in GF(prime(n)) if 2 is a quadratic residue (mod prime(n)) or GF(prime(n)^2) otherwise. Thus a(n) divides prime(n)-1 if prime(n) == 1 or 7 (mod 8), and a(n) divides prime(n)+1 if prime(n) == 3 or 5 (mod 8). - Robert Israel, Aug 28 2015

MAPLE

local p, r, G;

uses numtheory;

p:= ithprime(n);

if quadres(2, p)=1 then

else

fi

#20 by Robert Israel at Fri Aug 28 15:41:41 EDT 2015

MAPLE

f:= proc(n)

local p, r, G;

uses numtheory;

p:= ithprime(n);

if quadres(2, p)=1 then

r:= msqrt(2, p);

order(-3-2*r, p)

else

G:= GF(p, 2, r^2-2);

G:-order( G:-ConvertIn(-3-2*r));

fi

end proc:

2, seq(f(n), n=2..100); # Robert Israel, Aug 28 2015

STATUS

approved

editing

#19 by Charles R Greathouse IV at Tue Jun 16 11:20:51 EDT 2015

STATUS

editing

approved