A227782 Stufe of Q(sqrt(-n)): least number of squares which add to -1 in the field Q(sqrt(-n)).

1, 2, 2, 1, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 4, 2, 1, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 1, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2

Offset

1,2

Comments

If n >= 0 then Q(sqrt(n)) is formally real and its stufe is said to be infinite.

References

Ian G. Connell, The stufe of number fields, Mathematische Zeitschrift 124:1 (1972), pp. 20-22.

A. R. Rajwade, Squares, Cambridge Univ. Press, 1983.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Trygve Nagell, Sur la résolubilité de l'équation x^2 + y^2 + z^2 = 0 dans un corps quadratique, Acta Arithmetica 21:1 (1972), pp. 35-43.

A. R. Rajwade, A note on the stufe of quadratic fields, Indian J. Pure Appl. Math. 6:7 (1975), pp. 725-726.

Kazimierz Szymiczek, Note on a paper by T. Nagell, Acta Arithmetica 25:3 (1974), pp. 313f.

Wikipedia, Stufe (algebra)

Formula

a(n) = 1 if n is a square; a(n) = 4 if n is of the form 4^k(8m+7) for some m; a(n) = 2 otherwise.

Example

a(1) = a(4) = a(9) = 1 since Q(sqrt(-1)) = Q(sqrt(-4)) = Q(sqrt(-9)) = Q(i) has a square equal to -1: i^2 = -1.
a(3) = 2 since ((w+1)/2)^2 + ((w-1)/2)^2 = -1 where w = sqrt(-3).

Prog

(PARI) a(n)=n=core(n); if(n%8==7, 4, if(n==1, 1, 2))
(PARI) a(n)=n>>=(valuation(n, 2)\2*2); if(n%8==7, 4, 2-issquare(n))

Crossrefs

Cf. A227781, A227783.

Sequence in context: A038541 A070215 A071457 * A255771 A334590 A115034

Adjacent sequences: A227779 A227780 A227781 * A227783 A227784 A227785

Keyword

nonn,easy

Author

Charles R Greathouse IV, Jul 31 2013

Status

approved