The Collatz conjecture is a fascinating mathematical puzzle that's surprisingly simple to describe but incrediblly difficult to prove. Imagine a game where you start with any positive whole (counting) number. If it's even (divisble by 2), you divide it by 2. If it's odd, you multiply it by 3 and then add 1. Then you repeat this process with the new number you get. The conjecture states that no matter what number you start with, you'll always eventually reach 1[1].
For example, if you start with 10 the squence is: 10 → 5 → 16 → 8 → 4 → 2 → 1
This sequence is often called the "hailstone sequence" because the numbers bounce up and down like hailstones in a cloud before eventually falling to the ground (reaching 1)[1].
Lothar Collatz was a German mathematician who proposed this conjecture in 1937. He was known for his work in numerical analysis and graph theory. Collatz introduced the problem during his studies at the University of Hamburg, but it wasn't widely known until later years[2].
- 1937: Lothar Collatz proposes the conjecture.
- 1970s: The problem gains widespread attention in the mathematical community.
- 1986: Hungarian mathematician László Varga proves the conjecture for all numbers up to
$2^240$ (about a trillion). - 2019: Terence Tao makes significant progress, proving a weaker version of the conjecture.
- 2020: A breakthrough by Marijn Heule uses SAT solvers to verify the conjecture up to
$2^268$ (about 295 trillion)[3].
The Collatz conjecture is known by several names, reflecting its broad recognition in the mathematical community. Here are some of the most common alternative names:
-
$3n + 1$ Conjecture: This name highlights the rule applied to odd numbers in the sequence. - Syracuse Problem: Named after Syracuse University, where some notable work on the conjecture was conducted.
- Ulam Conjecture: Named after mathematician Stanisław Ulam, who popularised interest in this problem.
- Hailstone Sequence: This term describes the behaviuor of the numbers in the sequence, which rise and fall like hailstones before eventually settling at 1.
- Wondrous Numbers: Another less common name that reflects the curiosity and intrigue surrounding the conjecture.
- The longest known sequence before reaching 1 starts with 27, taking 111 steps.
- The conjecture has been tested for all numbers up to
$2^268$ , but a proof for all numbers remains elusive. - While it doesn't have direct practical applications, the Collatz conjecture has inspired algorithms in computer science and cryptography.
- The conjecture was featured in an episode of the TV show "Futurama" titled "The Prisoner of Benda."
- It's mentioned in the novel "Uncle Petros and Goldbach's Conjecture" by Apostolos Doxiadis.
- The conjecture has inspired various artistic works, including musical compositions based on the sequence of numbers.
The Collatz conjecture continues to fascinate mathematicians and puzzle enthusiasts alike, remaining one of the most intriguing unsolved problems in mathematics.
References:
- [1] https://simple.wikipedia.org/wiki/Collatz_conjecture
- [2] https://testbook.com/maths/collatz-conjecture
- [3] https://terrytao.files.wordpress.com/2020/02/collatz.pdf
- [4] https://github.com/StephenDsouza90/python-collatz
- [5] https://en.wikipedia.org/wiki/Collatz_conjecture
- [6] https://study.com/academy/lesson/history-of-the-collatz-conjecture.html
- [7] https://pubs.sciepub.com/ajams/9/3/5/index.html
- [8] https://vixra.org/pdf/2202.0064v1.pdf