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A246904
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Irregular triangular array: every periodic simple continued fraction CF represents a quadratic irrational (c + f*sqrt(d))/b, where b,c,f,d are integers and d is squarefree. Row n of this array shows the distinct values of d as CF ranges through the periodic continued fractions having period an n-tuple of 1s and 2s.
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4
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2, 5, 2, 3, 5, 2, 5, 10, 85, 2, 3, 5, 6, 30, 221, 2, 5, 13, 65, 145, 290, 533, 2813, 2, 3, 5, 10, 42, 85, 87, 105, 210, 455, 1365, 1517, 7565, 2, 5, 29, 58, 74, 149, 185, 442, 565, 901, 2026, 2117, 2210, 3973, 10001, 11026, 16133, 18229, 2, 3, 5, 6, 26, 30
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OFFSET
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1,1
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COMMENTS
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Is every squarefree positive integer in this array?
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LINKS
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EXAMPLE
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First 5 rows:
2 ... 5
2 ... 3 ... 5
2 ... 5 ... 10 .. 85
2 ... 3 ... 5 ... 6 ... 30 ... 221
2 ... 5 ... 13 .. 65 .. 145 .. 290 .. 533 .. 2813
The following list shows for n = 3 the 2^n purely periodic continued fractions, each followed by the number r it represents, the minimal polynomial a*x^2 + b*x + c of r, the discriminant, D = b^2 - 4*a*c, and the squarefree factor, d, of D.
[(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5, d = 5
[(1,1,2)] = sqrt(10)/4, -5 + 2 x^2, D = 40, d = 10
[(1,2,1)] = (2 + sqrt(10)/3, -2 - 4 x + 3 x^2, D = 10, d = 10
[(2,1,1)] = (1 + sqrt(85))/6, -7 - x + 3 x^2, D = 85, d = 85
[(1,2,2)] = (1 + sqrt(10)/3, -3 - 2 x + 3 x^2, D = 10, d = 10
[(2,1,2)] = (-1 + sqrt(85))/6, -7 + x + 3 x^2, D = 85, d = 85
[(2,2,2)] = (5 + sqrt(85))/10, -3 - 5 x + 5 x^2, D = 85, d = 85
[(2,2,2)] = sqrt(2), -2 + x^2, D = 8, d = 2
The distinct values of d are 2, 5, 10, 85, as in row 3.
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MATHEMATICA
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z = 6;
t[n_] := t[n] = Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 2}, n]]
u[n_] := u[n] = Table[MinimalPolynomial[t[k], x], {k, 1, n}]
d = Discriminant[u[z], x]; v[n_] := Table[{p, m} =
Transpose[FactorInteger[k]]; Times @@ (p^Mod[m, 2]), {k, d[[n]]}];
w = Table[Union[Table[v[n], {n, 1, z}][[n]]], {n, 1, z}];
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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