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A227782
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Stufe of Q(sqrt(-n)): least number of squares which add to -1 in the field Q(sqrt(-n)).
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2
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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
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OFFSET
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1,2
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COMMENTS
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If n >= 0 then Q(sqrt(n)) is formally real and its stufe is said to be infinite.
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REFERENCES
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Ian G. Connell, The stufe of number fields, Mathematische Zeitschrift 124:1 (1972), pp. 20-22.
A. R. Rajwade, Squares, Cambridge Univ. Press, 1983.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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