18022Collatz type functions using Legendre/ Jacobi symbols
- May 9, 2006Has 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
check its Legendre symbol, base y, a prime
If Legendre[x/y] = 0 then x = some function of x F(x1) such
If Legendre[x/y] = 1 then x = some other function F(x2) of x
If Legendre[x/y) = -1 then x = a third function F(x3) of x
Repeat, looking at the Legendre symbols of x to output x. 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]?
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