page_title: Collatz conjecture
section_title: Iterating on real or complex numbers
hierarchical_section_title: Collatz conjecture / Extensions to larger domains / Iterating on real or complex numbers
context_page_description: The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.
  The conjecture is named after Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. It is also known as the 3n + 1 problem, the 3n + 1 conjecture, the Ulam conjecture, Kakutani's problem, the Thwaites conjecture, Hasse's algorithm, or the Syracuse problem.
  The sequence of numbers involved is sometimes referred to as the hailstone sequence or hailstone numbers, or as wondrous numbers.
  Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." He also offered US$500 for its solution. Jeffrey Lagarias in 2010 claimed that based only on known information about this problem, "this is an extraordinarily difficult problem, completely out of reach of present day mathematics."
context_section_description: The Collatz map can be viewed as the restriction to the integers of the smooth real and complex map
  If the standard Collatz map defined above is optimized by replacing the relation 3n + 1 with the common substitute "shortcut" relation , it can be viewed as the restriction to the integers of the smooth real and complex map
  The point of view of iteration on the real line was investigated by Chamberland (1996),. 
  He showed that the conjecture does not hold for real numbers since there are infinitely many fixed points, as well as orbits escaping monotonically to infinity. He also showed that there is, at least, another attracting cycle: 1.1925 → 2.1386.
  On the complex plane, it was investigated by Letherman, Schleicher, and Wood (1999).
  Most points of the plane diverge to infinity, as seen in blue on the illustration below. The boundary between diverging and non-diverging regions show a fractal pattern called the "Collatz fractal".
caption: Cobweb plot of the orbit 10-5-8-4-2-1-2-1-2-1-etc. in the real extension of the Collatz map (optimized by replacing "3n + 1" with "3n + 1/2")
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