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18022Collatz type functions using Legendre/ Jacobi symbols

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  • Robert
    May 9, 2006
      Has anyone looked at "Collatz-type" functions, using functions such as
      the Jacobi/ Legendre symbols?

      The function would look something like

      Start with an integer x[1]
      check its Legendre symbol, base y, a prime
      If Legendre[x[1]/y] = 0 then x[2] = some function of x[1] F(x1) such
      as x[1]+1
      If Legendre[x[1]/y] = 1 then x[2] = some other function F(x2) of x[1]
      If Legendre[x[1]/y) = -1 then x[2] = a third function F(x3) of x[1]

      Repeat, looking at the Legendre symbols of x[2] to output x[3]. Then
      stop at x[n], either a repeat of a previous x[] (loop) or x[n-1]
      (constant future values, preferably 1).

      There does not appear to be a constant definition of the Legendre and
      Jacobi Symbols, some sources, such as Zzmath for Excel, interpet
      Jacobi[2/2] as 1, whereas Maple shows this as 0. MathWorld has a
      statement that the Legendre symbol definiation is sometimes
      generalised such that Legendre[a/p]=0 when a|p

      What would be interesting choices of F[x1],F[x2] and F[x3]?


      Robert Smith
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