18030Re: [PrimeNumbers] Collatz type functions using Legendre/ Jacobi symbols
- May 10, 2006Robert <rw.smith@...> wrote: 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
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]?
No I haven't, but I' ve always wondered why 3n+1, in the case of odd numbers and n/2 in the case of even numbers, cannot simply be collapsed into n+((n+1)/2)) (in the case of odd numbers), thus jumping a step.
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