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

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

Author

N. J. A. Sloane

Keywords

• nonn

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)

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.
• Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.
• 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.

Crossrefs

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

Programs

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 *)

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

Extensions

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

More terms from Jud McCranie, Mar 18 2001

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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