collatz_polynomial

collatz_polynomial, a MATLAB code which implements the Collatz polynomial iteration, a polynomial analog of the numerical iteration that is also known as the 3n+1 conjecture or the hailstone sequence.

Let p0(x) be a polynomial with integer coefficients mod 2. Then define the next Collatz polynomial p1(x) as follows:

1. if p0(x) is divisible by x, p1(x) = p0(x) / x.
2. Otherwise, p1(x) = p0(x) * ( x + 1 ) + 1 mod 2.

This transformation can be repeated, generating a Collatz polynomial sequence. For all starting polynomials checked so far, the sequence reaches the polynomial p(x) = 1, at which point the convention is to halt. (There is also the exceptional polynomial p(x)=0.) It is interesting to investigate the number of steps required to drive a particular polynomial to 1, and to find patterns in this behavior.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

collatz_polynomial is available in a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

collatz_polynomial_test

collatz, a MATLAB code which computes and analyzes the Collatz or hailstone or 3n+1 sequence;

matlab_polynomial, a MATLAB code which analyzes a variety of polynomial families, returning the polynomial values, coefficients, derivatives, integrals, roots, or other information.

Reference:

1. John D Cook,
   Polynomial analog of the Collatz conjecture,
   27 July 2021,
   https://www.johndcook.com/blog/2021/07/27/polynomial-collatz/

Source Code:

• collatz_polynomial_next.m, returns the next Collatz polynomial.
• collatz_polynomial_sequence.m, prints a sequence of Collatz polynomials for a given starting polynomial.
• i4vec_is_binary.m, is TRUE if all entries of a vector are 0 or 1.
• polynomial_degree.m, returns the degree of a polynomial.
• polynomial_print.m, prints a polynomial with integer coefficients.