1,1

In other words, if p is the n-th prime == 1 mod 4, a(n) is the smallest positive integer k such that k^2 + 1 == 0 mod p.

The 4th roots of unity mod p, where p = n-th prime == 1 mod 4, are +1, -1, a(n) and p-a(n).

Related to Stormer numbers.

Comment from Igor Shparlinski, Mar 12 2007 (writing to the Number Theory List): Results about the distribution of roots (for arbitrary quadratic polynomials) are given by W. Duke, J. B. Friedlander and H. Iwaniec and A. Toth.

Comment from Emmanuel Kowalski, Mar 12 2007 (writing to the Number Theory List): It is known (Duke, Friedlander, Iwaniec, Annals of Math. 141 (1995)) that the fractional part of a(n)/p(n) is equidistributed in [0,1/2] for p(n)<X and X going to infinity. So a positive proportion of p have a between xp and yp for 0<x<y<1/2, but equidistribution in smaller sets is not known.

Contribution from Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008: (Start)

If we take 4 numbers : 1, A002314(n), A152676(n), A152680(n) then

multiplication table modulo A002144(n) is isomorphc to the Latin square:

1 2 3 4

2 4 1 3

3 1 4 2

4 3 2 1

and isomorphic to the multiplication table of {1, I, -I, -1} where I is Sqrt[ -1],

A152680(n) is isomorphic to -1, A002314(n) with I or -I and A152676(n) vice versa -I or I.

1, A002314(n), A152676(n), A152680(n) are subfield of Galois Field [A002144(n)]. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

W. Duke, J. B. Friedlander and H. Iwaniec, Equidistribution of roots of a quadratic congruence to prime moduli, Annals of Math, 141 (1995), 423-441.

J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.

A. Toth, Roots of quadratic congruences, Intern. Math. Research Notices, 2000 (2000), 719-739.

T. D. Noe, Table of n, a(n) for n=1..1000

f:=proc(n) local i, j, k; for i from 1 to (n-1)/2 do if i^2 +1 mod n = 0 then RETURN(i); fi od: -1; end;

t1:=[]; M:=40; for n from 1 to M do q:=ithprime(n); if q mod 4 = 1 then t1:=[op(t1), f(q)]; fi; od: t1;

aa = {}; Do[If[Mod[Prime[n], 4] == 1, k = 1; While[ ! Mod[k^2 + 1, Prime[n]] == 0, k++ ]; AppendTo[aa, k]], {n, 1, 100}]; aa [From Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008]

Cf. A002313, A005528, A047818.

A002144, A152676, A152680 [From Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008]

Sequence in context: A072403 A010078 A074639 this_sequence A094471 A126356 A121274

Adjacent sequences: A002311 A002312 A002313 this_sequence A002315 A002316 A002317

nonn

N. J. A. Sloane (njas(AT)research.att.com).

Better description from Tony Davie (ad(AT)dcs.st-and.ac.uk), Feb 07 2001

More terms from Jud McCranie (j.mccranie(AT)comcast.net), Mar 18, 2001.