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A002314
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Minimal integer square root of -1 modulo p(n), where p(n) = n-th prime of form 4k+1.
(Formerly M1314 N0503)
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15
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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.
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)
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REFERENCES
| 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.
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).
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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
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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;
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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 [From Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008]
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CROSSREFS
| Cf. A002313, A005528, A047818.
A002144, A152676, A152680 [From Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008]
Sequence in context: A010078 A074639 * A177979 A094471 A126356 A121274
Adjacent sequences: A002311 A002312 A002313 * A002315 A002316 A002317
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Better description from Tony Davie (ad(AT)dcs.st-and.ac.uk), Feb 07 2001
More terms from Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Mar 18, 2001.
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