A002314 Minimal integer square root of -1 modulo p, where p is the n-th prime of the form 4k+1. (Formerly M1314 N0503)

2, 5, 4, 12, 6, 9, 23, 11, 27, 34, 22, 10, 33, 15, 37, 44, 28, 80, 19, 81, 14, 107, 89, 64, 16, 82, 60, 53, 138, 25, 114, 148, 136, 42, 104, 115, 63, 20, 143, 29, 179, 67, 109, 48, 208, 235, 52, 118, 86, 24, 77, 125, 35, 194, 154, 149, 106, 58, 26, 135, 96, 353, 87, 39

Offset

1,1

Comments

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.

From Artur Jasinski, Dec 10 2008: (Start)

If we take the four numbers 1, A002314(n), A152676(n), and A152680(n), then their multiplication table modulo A002144(n) is isomorphic 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=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)

It is found empirically that the solutions of the Diophantine equation X^4 + Y^2 == 0 (mod P) (where P is a prime of the form P=4k+1) are integer points on parabolas Y = (+-(X^2 - P*X) + P*i)/C(P) where C(P) is the term corresponding to a prime P in this sequence. - Seppo Mustonen, Sep 22 2020

References

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

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

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, pp. 1-21.

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.

Seppo Mustonen, Roots of Diophantine equations mod(X^4+Y^2,P)=0, 2020

Seppo Mustonen, Roots of Diophantine equations mod(X^4+Y^2,P)=0, Youtube video, 2020

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.

Maple

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;

Mathematica

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 (* Artur Jasinski, Dec 10 2008 *)

Prog

(PARI) first_N_terms(N) = my(v=vector(N), i=0); forprime(p=5, oo, if(p%4==1, i++; v[i] = lift(sqrt(Mod(-1, p)))); if(i==N, break())); v \\ Jianing Song, Apr 17 2021

Crossrefs

Cf. A002313, A005528, A047818, A002144, A152676, A152680. Subsequence of A057756.

Sequence in context: A357983 A338459 A343035 * A177979 A362707 A094471

Adjacent sequences: A002311 A002312 A002313 * A002315 A002316 A002317

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

More terms from Jud McCranie, Mar 18 2001

Status

approved