Paul Erdős’s Contributions to Mathematics
Photo by Oliver Helbig/Getty Images
In an astonishing development, just a week after an AI system disproved a long-standing mathematical conjecture, another enduring conjecture—one that is over fifty years old—has also fallen, this time due to entirely human effort.
Recently, OpenAI’s advanced model refuted the significant unit distance problem, originally posed by Hungarian mathematician Paul Erdős. This problem, regarded by Erdős as his “most important contribution to geometry,” explores the maximum number of equal-distance connections that can be drawn between points in a plane.
Erdős proposed a maximum limit for this value, which many scholars believed to be accurate. However, AI’s findings suggested that this figure could be significantly higher. By employing intricate methods from algebraic number theory, mathematicians could devise high-dimensional structures that differ from previous human designs, resulting in unprecedented surprises within the mathematical community.
Less than a week later, Professor Thomas Bloom and his team at the University of Manchester leveraged a similar approach to invalidate the well-known sum-product conjecture, first introduced by Erdős in 1976.
“I was amazed because I had been pondering this issue for a while,” Bloom stated. His team recognized the algebraic techniques employed by OpenAI’s AI and applied them to the sum-product conjecture. “Once you see a possibility, it drives you to make it happen,” he explained.
Erdős’s Wasumi conjecture posits that when summing or multiplying a set of numbers, at least one of the resulting sets must vastly exceed the original in size, while simultaneously, both cannot be minimized equally. For example, the multiplication of numbers from 1 to 5 yields a larger set than their sum due to overlaps like 2 + 3 and 1 + 4. If we analyze a set like 1, 2, 4, 8, 16, etc., the summed set is larger since the product simply yields different powers of 2.
Erdős established a standard for the minimal size of the larger set generated from summation and multiplication, which he believed would hold true for all numerical sets. However, Bloom and his team adapted the high-dimensional method to find instances where both the summation and multiplication were smaller than Erdős anticipated. Rather than using a geometric progression, such as powers of 2, they discovered that various dimensional progressions could yield startling results with fewer unique totals than previously thought.
“What astonished me was how straightforward it was,” Bloom remarked. “The underlying structure is simple, yet now I better grasp the underlying reasons.” He believes [Erdős’s conjecture] has indeed failed, but also sees potential implications for multiple related mathematical issues.
“Mathematics is competitive,” said Mischa Rudnev from the University of Bristol. “As soon as a fresh idea materializes, many rush to find further applications, and these enthusiasts are typically brilliant and swift.”
Rudnev noted that Erdős’ initial belief was that this conjecture mainly applies to integers, a notion that still holds as the new sets Bloom’s team created utilized increasingly complex number systems. Bloom concurs that while it remains valid for integers, “significant work is yet to come, and the intricacies are not fully understood.”
Bloom highlights the key takeaway from this proof: problems traditionally viewed as geometric, such as powers of two, can be approached with number theory tools. “This opens these problems to a new audience. The algebraic number theorists hadn’t shown much interest in these issues previously.”
Topics:
- Artificial Intelligence/
- Mathematics
Source: www.newscientist.com
