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Farey Tree segregation of n/29 into QR and NQR's

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  • purushaz
    ... Cataloging rational fractions with 29 denominator, those fractions with numerators a,b,c...being quadratic residues..are on altogether different rows of
    Message 1 of 1 , Sep 30, 2002
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      Cataloging rational fractions with "29" denominator, those fractions
      with numerators a,b,c...being quadratic residues..are on altogether
      different rows of the Infinite Farey tree than non-QR's.

      Background...get continued fraction form of the fraction and add up
      the terms...e.g. 10/29 is on the 12th row since 10/29 = [2, 1, 9].

      1/29 and 28/29 are both in the 29th row (1 and 28 both Quadratic
      Residues of 29). I don't know if any primes higher than 29 have the
      same property of mutually exclusive rows in the Infinite Farey Tree,
      QR and non-QR numerators.(mainly because it's tedious by hand...maybe
      somebody could come up with a pgm). At any rate, here's a further
      breakdown:

      As to non- Quadratic residues:
      Row 16: numerators are 2, 15, 27
      Row 8, : 8, 11, 18, 21
      Row 12: 3, 10, 19, 26
      Row 8: 12, 17
      Row 16: 14, 15, 27.

      As to Quadratic Residues:
      Row 10: 5, 6, 24, 23
      Row 11: 4, 7, 22, 25
      Row 9: 9, 16, 20, 13

      ...Gary Adamson
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