Pick a number; if it is even, divide it by 2; if odd, triple it and add 1. Now repeat the process, each time halving or else tripling and adding 1. Here is a surprise: no matter what number you pick, you will eventually arrive at 1. Let’s try 6: it is even, so we halve it to get 3, which is odd so we triple and add 1 to get 10. Thereafter we have 5, 16, 8, 4, 2 and 1.
From then on, the value cycles from 1 to 4 to 2 and back to 1 again, for ever. Numerical checks have shown that all numbers up to one hundred million million million reach the 1-4-2-1 cycle.
The process of producing the sequence is encapsulated in the mnemonic HOTPO, for "Half, or Triple Plus One". A conjecture was made in 1937 by German mathematician Lothar Collatz that, no matter what value we start from, the sequence always reaches 1 after a finite number of steps. This is the Collatz Conjecture. It is difficult to doubt the truth of the conjecture, but mathematicians have been unable to prove it.
The renowned Hungarian mathematician Paul Erdös said of the Collatz Conjecture, “Mathematics may not be ready for such problems”. A full resolution seems to be beyond the reach of current methods. However, progress is possible if we seek results that hold for most numbers.
The word "most" has a precise technical meaning here, based on logarithmic density. About a year ago Australian-American mathematician Terence Tao gave a proof that "almost all Collatz orbits attain almost bounded values". This makes it very improbable that a counter-example to the conjecture exists, but the full problem remains open.
Tao, born in Adelaide Australia, showed remarkable mathematical abilities from an early age. He first competed in the International Mathematical Olympiad aged 10, the youngest participant ever, and won a gold medal at the age of 13. Tao was awarded a PhD at Princeton University at the age of 21 – he is now 45.
A list of Tao’s research interests would require a long paragraph filled with technical terms. His research has spanned a remarkable breadth and he has made fundamental contributions in many areas of pure and applied mathematics ranging from number theory to the Navier-Stokes equations, which govern atmospheric motions and other fluid flows.
Tao was awarded a Fields Medal in 2006. This honour, generally regarded as the Nobel Prize for mathematics, is awarded at the International Congress of the International Mathematical Union every four years.
The award citation for Tao listed four distinct areas to which he has made singular contributions. Tao has received several other major awards, and has produced more than 350 papers and 17 books. He has been at the University of California since 1999 and currently holds the James and Carol Collins chair in mathematics there.
The 2020 Hamilton Lecture, organised by the Royal Irish Academy, will be delivered by Tao on Friday, October 16th at 4pm, by video link from Los Angeles. His lecture, The Cosmic Distance Ladder, will provide some answers to questions such as "How do we know the distances from the Earth to the sun and moon, from the sun to the other planets, and from the sun to other stars and distant galaxies?" This is a free event but booking is essential at ria.ie.
Peter Lynch is emeritus professor at UCD school of mathematics and statistics. He blogs at thatsmaths.com