Big Advance on Simple-Sounding Math Problem Was a Century in the Making
Hector Pasten, a Chilean mathematician, advanced number theory by proving that the sequence n² + 1 always contains a large prime factor, impacting the abc conjecture and improving 1930s results.
Read original article
Hector Pasten, a mathematician from Chile, has made significant progress on a long-standing problem in number theory related to the sequence of numbers defined by n² + 1. This sequence has intrigued mathematicians for over a century due to its complex relationship between addition and multiplication. Pasten's breakthrough came unexpectedly while he was procrastinating on writing an exam, leading him to explore the prime factors of the sequence. He successfully demonstrated that these numbers must always have at least one large prime factor by embedding information about the prime factors into an elliptic curve equation. His proof, which was accepted by the journal Inventiones Mathematicae, marks a notable advancement in understanding the growth of prime factors in this sequence, improving upon results established in the 1930s. Additionally, Pasten's methods have implications for the abc conjecture, a famous unsolved problem in mathematics that also examines the interplay between addition and multiplication. His work is seen as a promising new direction in a field that has seen little progress on these questions for decades.
- Hector Pasten solved a long-standing problem in number theory related to the sequence n² + 1.
- His proof shows that numbers in this sequence must have at least one large prime factor.
- Pasten's work improves upon results from the 1930s regarding the growth of prime factors.
- His techniques also contribute to the abc conjecture, a significant unsolved problem in mathematics.
- The proof was accepted by Inventiones Mathematicae in a notably short time frame.
Related
The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved
Mathematicians Larry Guth and James Maynard made progress on the Riemann Hypothesis, a key problem in prime number distribution. Their work offers insights and techniques for potential breakthroughs in mathematics.
What Is the Riemann Hypothesis and Why Do People Want to Solve It
The Riemann Hypothesis, a century-old unsolved problem in mathematics, involves complex analysis and prime numbers' distribution along the number line, attracting global attention for its implications and challenges to mathematicians.
'Sensational' Proof Delivers New Insights into Prime Numbers
Mathematicians James Maynard and Larry Guth set new limits on exceptions to the Riemann hypothesis, improving prime number approximations. Their innovative approach highlights interdisciplinary collaboration's importance in solving mathematical problems.
Fibonacci Partial Sums Tricks
The paper discusses a mathematical trick for summing Fibonacci-like sequences, identifies the largest Fibonacci divisor of sums, and generalizes the method for Pell-like sequences, excluding Jacobsthal-like sequences.
A new rare high-rank elliptic curve, and an orchard of Diophantine equations
Recent developments in mathematics include Bogdan Grechuk's book on systematically solving Diophantine equations and the identification of a high-rank elliptic curve with 29 rational solutions, raising significant open problems.
3 comments> “Mathematics is not just about proving theorems — it’s about a way to interact with reality, maybe.”
This one I like it because in the current trend of trying to achieve theorem proving in AI only looking at formal systems, people rarely mention this.
And this one:
> Just what will emerge from those explorations is hard to foretell. “That’s the problem with originality,” Granville said. But “he’s definitely got something pretty cool.”
When has that been a "problem" with originality? Hahah but I understand what he means.
> another asks whether there are infinitely many pairs of primes that differ by only 2, such as 11 and 13
Is it that many questions were successfully dis/proved and so were left with some that seem arbitrary? Or is there something special about the particular questions mathematicians focus on that a layperson has no hope of appreciating?
My best guess is questions like the one above may not have any immediate utility, but could at any time (for hundreds of years) become vital to solving some important problem that generates huge value through its applications.
Related
The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved
Mathematicians Larry Guth and James Maynard made progress on the Riemann Hypothesis, a key problem in prime number distribution. Their work offers insights and techniques for potential breakthroughs in mathematics.
What Is the Riemann Hypothesis and Why Do People Want to Solve It
The Riemann Hypothesis, a century-old unsolved problem in mathematics, involves complex analysis and prime numbers' distribution along the number line, attracting global attention for its implications and challenges to mathematicians.
'Sensational' Proof Delivers New Insights into Prime Numbers
Mathematicians James Maynard and Larry Guth set new limits on exceptions to the Riemann hypothesis, improving prime number approximations. Their innovative approach highlights interdisciplinary collaboration's importance in solving mathematical problems.
Fibonacci Partial Sums Tricks
The paper discusses a mathematical trick for summing Fibonacci-like sequences, identifies the largest Fibonacci divisor of sums, and generalizes the method for Pell-like sequences, excluding Jacobsthal-like sequences.
A new rare high-rank elliptic curve, and an orchard of Diophantine equations
Recent developments in mathematics include Bogdan Grechuk's book on systematically solving Diophantine equations and the identification of a high-rank elliptic curve with 29 rational solutions, raising significant open problems.