Who Invented the 3x + 1 Problem? An Epic, 80-Year Math Mystery
For mathematicians, no math riddle has proven more tantalizing than the 3x + 1 problem. This deceptively simple iterative function was first proposed in 1937 by German mathematician Lothar Collatz, and over 80 years later, the problem remains unsolved. Why has the 3x + 1 conjecture captivated generations of math lovers? Let‘s dive into the problem‘s origins and why it continues to maintain an air of intrigue after all this time.
The Enigmatic Lothar Collatz
Born in 1910, Lothar Collatz grew up in Germany and developed a passion for mathematics from a young age. He studied a broad range of mathematical fields at university, including number theory, algebra, and probability.
Appointed as a lecturer at the University of Hamburg in 1932, Collatz focused his research on iterative function sequences. Pure mathematics was progressing swiftly in the early 20th century, with entire new vistas around abstraction being explored.
It was against this backdrop that, in 1937, Collatz cryptically presented a research problem concerning the following function during a lecture:
f(n) = n/2 if n is even
3n + 1 if n is oddGiven any positive integer as input, Collatz claimed that repeatedly applying this function would always eventually reach 1, no matter how large the starting number.
Despite the simplicity of the statement, Collatz did not provide a proof. He merely posed it as a conjecture, which over the next 80+ years would capture the fascination of professional and hobbyist mathematicians alike.
Why This Captivated the Math World
At first glance, the 3x + 1 function looks almost trivial – if n is even, halve it, if odd – multiply by 3 and add 1. What makes this problem so mysteriously hard to crack?
For starters, the cycles produced by iterating this function appear chaotic, especially as the input n grows larger. Traditional number theory arguments that mathematicians attempted did not make progress. It seemed to span multiple domains – from recreational math to abstract algebra – without fitting perfectly into any.
As computers advanced, mathematicians were able to verify Collatz‘s conjecture for increasingly larger numbers, but a general proof remained unattainable.
The Simpson‘s writer Simon Singh best captures why the problem maintains an air of intrigue even today:
"It appeals to the mind of a child and yet it goes beyond the greatest minds of history…The sheer simplicity of the problem and the statement, the contradiction of that by the fact that it‘s unsolved, really draws in a math enthusiast."
Beyond just the math world, it resonated deeply across fields – from computer science to cyber security. Hacking pioneer Kevin Mitnick describes it as the perfect "cipher" problem for encrypting communication since proving validity seems impossibly hard.
Unsuccessful Attempts Across Decades
Between then and now, both amateurs and professionals have tried endless strategies to tackle the infamous 3x + 1, with little success.
In 1954, Kurt Heun became the first to examine and attempt to disprove the conjecture computationally by checking numbers up to 105 = 100,000.
By the 1970s and 80s, computers enabled verifying 3x + 1 for much larger numbers, even billions in some cases. But a theoretical proof remained elusive as ever.
In 2022, a preprint paper on arXiv received lots of attention by claiming to prove Collatz using a 57 page computer assisted proof. However, within days other mathematicians identified logical gaps that rendered it inconclusive.
This demonstrates the challenge and pitfalls with trying to establish computational certainty around an infinite set of numbers.
New Approaches Using Heuristics & ML
In the past decade, new attempts have risen to solve Collatz using heuristic algorithms, probabilistic reasoning, and machine learning models.
In 2013, mathematician Jeffrey Lagarias stated in an interview that "New ideas are needed to make progress on the Collatz conjecture".
This sparked creative approaches from non traditional angles to tackle 3x + 1:
In 2019, an attempt using genetic algorithms and neural nets to evolve heuristic based proofs was published in the Journal of Computational Science. It made interesting initial progress but couldn‘t fully prove the conjecture.
In September 2022, a preprint paper on optimization based proof methods was posted on arXiv, though its accuracy is still under examination.
Such cross-domain techniques demonstrate renewed hope of finally designing an innovative approach to crack Collatz.
Will We See A Proof In Our Lifetimes?
In 2006, famous mathematician Terence Tao stated that "mathematics is not yet advanced enough to solve this problem". So when, if ever, will we finally put this 80+ year old riddle to bed?
On one hand, the problem‘s unpredictability makes believing we‘ll see a proof within our lifetime feel optimistic.
On the other hand, modern computers enable us to check and analyze massive sets of integers iteratively in the hunt for patterns. Coupled with creative heuristics using AI/ML, maybe we will see an ingenious proof emerge soon that catches us all by surprise!
As Tao himself said, "The key is to introduce a new idea that did not occur to anyone before."
Until mathematically proven either way, the mysterious 3x + 1 conjecture will likely continue intriguing math enthusiasts for generations to come as we ponder the genius yet enigmatic Lothar Collatz – the man who started it all.
