Collatz Problem

Problem's Description

The Collatz conjecture is among the most renowned unsolved problems in mathematics. It postulates whether iteratively applying two elementary arithmetic operations will inevitably lead every positive integer to 1.

It revolves around sequences of integers where each term is derived from the preceding one by the following rules:

if the previous term is even, the next term is half of it, and if the previous term is odd, the next term is three times the previous term plus one.

If iterating

$$ a_n = \begin{cases} \frac{1}{2} a_{n-1} & \text{for } a_{n-1} \text{ even} \\ 3 a_{n-1} + 1 & \text{for } a_{n-1} \text{ odd} \end{cases} $$

always returns to 1 for positive $ a_0 $. (If negative numbers are included, there are four known cycles (excluding the trivial 0 cycle): (4, 2, 1), (-2, -1), (-5, -14, -7, -20, -10), and (-17, -50, -25, -74, -37, -110, -55, -164, -82, -41, -122, -61, -182, -91, -272, -136, -68, -34).)

The conjecture posits that all such sequences ultimately reach 1, regardless of the initial positive integer chosen. While it has been validated for all attempted positive integers, even up to an exceedingly large number like $2.95 \times 10^{20}$, a general proof remains elusive.

Named after mathematician Lothar Collatz, who proposed the concept in 1937, two years following his doctoral degree. These sequences are alternatively known as hailstone sequences, hailstone numbers, or hailstone numerals due to their characteristic pattern resembling the descent and ascent of hailstones in a cloud, or sometimes as wondrous numbers.

Paul Erdős remarked on the Collatz conjecture: "Mathematics may not be ready for such problems." Jeffrey Lagarias, in 2010, described the Collatz conjecture as "an extraordinarily challenging problem, currently beyond the capabilities of contemporary mathematics."

1. Chapter "Lothar Collatz"

   • E.F.,
   • J.J.;,
   • O'Connor,
   • Robertson

   St Andrews University School of Mathematics and Statistics, Scotland.