cp's OEIS Frontend

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

%I A002314 M1314 N0503 #60 Oct 09 2024 04:33:03
%S A002314 2,5,4,12,6,9,23,11,27,34,22,10,33,15,37,44,28,80,19,81,14,107,89,64,
%T A002314 16,82,60,53,138,25,114,148,136,42,104,115,63,20,143,29,179,67,109,48,
%U A002314 208,235,52,118,86,24,77,125,35,194,154,149,106,58,26,135,96,353,87,39
%N A002314 Minimal integer square root of -1 modulo p, where p is the n-th prime of the form 4k+1.
%C A002314 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).
%C A002314 The 4th roots of unity mod p, where p = n-th prime == 1 (mod 4), are +1, -1, a(n) and p-a(n).
%C A002314 Related to Stormer numbers.
%C A002314 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.
%C A002314 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.
%C A002314 From _Artur Jasinski_, Dec 10 2008: (Start)
%C A002314 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
%C A002314     1 2 3 4
%C A002314     2 4 1 3
%C A002314     3 1 4 2
%C A002314     4 3 2 1
%C A002314 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.
%C A002314 1, A002314(n), A152676(n), A152680(n) are subfield of Galois Field [A002144(n)]. (End)
%C A002314 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
%D A002314 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002314 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002314 Jon E. Schoenfield, <a href="/A002314/b002314.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A002314 A. J. C. Cunningham, <a href="https://archive.org/details/binomialfactoris01cunn/page/1/mode/1up">Binomial Factorisations</a>, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, pp. 1-21.
%H A002314 W. Duke, J. B. Friedlander and H. Iwaniec, <a href="https://www.jstor.org/stable/2118527">Equidistribution of roots of a quadratic congruence to prime moduli</a>, Annals of Math, 141 (1995), 423-441.
%H A002314 Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, <a href="https://doi.org/10.7546/nntdm.2024.30.3.516-529">The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences</a>, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.
%H A002314 Seppo Mustonen, <a href="https://www.survo.fi/demos/index.html#ex138">Roots of Diophantine equations mod(X^4+Y^2,P)=0</a>, 2020
%H A002314 Seppo Mustonen, <a href="https://youtu.be/FS1dGHf5RPk">Roots of Diophantine equations mod(X^4+Y^2,P)=0</a>, Youtube video, 2020
%H A002314 J. Todd, <a href="https://www.jstor.org/stable/2305526">A problem on arc tangent relations</a>, Amer. Math. Monthly, 56 (1949), 517-528.
%H A002314 A. Toth, <a href="https://doi.org/10.1155/S1073792800000404">Roots of quadratic congruences</a>, Intern. Math. Research Notices, 2000 (2000), 719-739.
%p A002314 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;
%p A002314 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;
%t A002314 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 *)
%o A002314 (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
%Y A002314 Cf. A002313, A005528, A047818, A002144, A152676, A152680. Subsequence of A057756.
%K A002314 nonn
%O A002314 1,1
%A A002314 _N. J. A. Sloane_
%E A002314 Better description from Tony Davie (ad(AT)dcs.st-and.ac.uk), Feb 07 2001
%E A002314 More terms from _Jud McCranie_, Mar 18 2001