The solution to the problem forgotten by humans has been rediscovered by GPT-5 Pro!
The incident focuses on Erdös Problem #339 , one of nearly a thousand problems posed or paraphrased by renowned mathematician Paul Erdös and listed on the website erdosproblems.com. The website keeps track of the current status of each problem, with about a third solved and the majority still to be solved.
Previously, this problem was marked as "unsolved" and was a difficult mathematical problem to be solved. Many people are still continuing to study and explore it.
It wasn't until recently that someone used GPT-5 Pro to search and discovered that the problem had actually been solved in 2003 .
What is particularly noteworthy is that GPT-5 Pro directly located the key document using only the image of Erdős question #339.
After OpenAI researcher Sebastien Bubeck shared this matter, it immediately attracted the attention of a large number of netizens.
By the way, one of Terence Tao's famous achievements is that he broke through the "Erdős discrepancy problem", a conjecture that has plagued the mathematical community for decades, by using the tool of "ergodic theory".
Problem Details
Specifically, Erdős Problem #339 is a classic problem in number theory in the direction of additive basis, which can be expressed as:
Let A⊆N be a basis of order r (i.e., every sufficiently large integer can be expressed as the sum of r elements in A). Then, does the set of integers that can be expressed as the sum of exactly r distinct elements in A necessarily have a positive lower density?
In addition, Erdős and Graham also raised a related question: If the set of integers that can be expressed as the sum of r elements in A has a positive upper density, then does the set of integers that can be expressed as the sum of exactly r different elements in A also have a positive upper density?
Before GPT-5 Pro discovered that this problem had been solved, netizens had a series of discussions on this issue on the website.
Starting from the famous Waring's Problem, netizen Adenwalla pointed out that almost all integers can be expressed as the sum of up to 15 fourth powers, but there are still infinitely many integers that require 16 fourth powers, that is, G(4)=16 but G₁(4)=15.
This leads to thinking: does this mean that the lower density conclusion in the additive basis problem may not hold?
Soon, Woett, Boris Alexeev and others pointed out that the example in the Waring problem was a case where "elements are allowed to be repeated", while Erdős problem #339 required that "elements are different from each other", so this example could not constitute a counterexample, and the conditions of the original problem were more stringent.
The discussion then deepened further.
Zach Hunter sought to explore the density stability of additive bases at different scales, while Woett proposed some specific set constructions as counterexamples to the proposed disconfirmation. The two sides debated concepts such as "distinct elements," "lower density," and "bounded doubling."
In the end, they found that although these constructions could create examples with sparse or even exponential gaps in the size of the sum set, they still could not make the lower density of "the set of integers that can be expressed as the sum of exactly r different elements" truly approach zero. In other words, these counterexample constructions did not successfully disprove the proposition.
Just when netizens were arguing with each other and there was still controversy over whether the question was valid.
Msawhney reminds everyone that this problem was actually solved as early as 2003.
The core basis is the paper "A proof of two Erdos' conjectures on restricted addition and further results" by Hegyvari, Hennecart, and Plagne, published in "J. reine angew. Math." (i.e. "Crelle"), Volume 560, pages 199-220.
Theorem 4 directly constitutes the solution to this problem.
It was GPT-5 Pro that found the answer. It accurately located the document based solely on a screenshot of the question.
About Paul Erdös
Paul Erdős was one of the most outstanding and prolific mathematicians of the 20th century, known for his significant contributions to number theory, combinatorics, graph theory, probability theory and other fields.
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He published nearly 1,500 papers in his lifetime and conducted research with more than 500 collaborators . His extensive spirit of cooperation gave rise to the concept of " Erdős number " in the mathematics community. This number has become an "honor indicator" to measure the closeness of a mathematician's academic connection with Erdős.
He was born in Budapest, Hungary in 1913. At the age of 4, he could already do multi-digit multiplication in his head. At the age of 10, he taught himself the entire high school mathematics curriculum and began studying number theory.
In 1934, 21-year-old Erdös received his doctorate from the University of Budapest, and then began to "drift" due to the influence of war and other factors.
He has no fixed position and lives on speaking fees, bonuses and support from friends. He always carries a suitcase and travels to universities and mathematicians' homes around the world, collaborating with his colleagues on research and discussing problems, changing places every few weeks on average.
Throughout his life, Erdös was known for his problem-driven approach to research . Rather than pursuing systematic theories, he continually raised and solved interesting problems. Hundreds of his conjectures remain at the forefront of mathematics today.
Number theory was Erdös's most profound and fruitful field. His work directly advanced the development of number theory in the 20th century, particularly in the areas of prime number distribution and additive number theory. For example, he and the Norwegian mathematician Atle Selberg used elementary methods to prove the prime number theorem, a result that astonished the mathematical community.
Erdös is also one of the founders of Ramsey number research. He introduced probability theory into combinatorial number theory and gave an estimate of the lower bound of Ramsey numbers.
The famous "Erdos difference problem" he proposed can be traced back to the 1930s and 1940s.
The content is that given an infinite sequence of +1 and -1 (such as (1, -1, 1, -1,…)), define the "partial sum of the first n terms" as S (n), then the "difference" refers to the maximum absolute value of all partial sums.
Erdös conjectured that for any such sequence, the difference will increase infinitely as n increases (i.e., there is no infinite ±1 sequence with "bounded difference").
This seemingly simple problem spans number theory, combinatorics, and harmonic analysis, becoming one of the most famous unsolved conjectures of the 20th century. It wasn't until 2015 that mathematician Terence Tao achieved a partial breakthrough in solving the conjecture by introducing a tool called ergodic theory.
Even in his last few years, Erdős continued to study mathematics and write papers. In 1996, he died of a heart attack while attending an academic conference in Warsaw, Poland, at the age of 83.
In 2024, British mathematician Thomas Bloom launched a website dedicated to the study of Erdős's problem.
One More Thing
Paata Ivanisvili, a mathematics professor at the University of California, Irvine, also tweeted that GPT-5Pro performed well in identifying serious flaws in published papers.
Five years ago, I spent days researching this paper and found a vulnerability that the authors later confirmed. GPT-5 Pro found the same vulnerability in just 18 minutes, along with several additional minor issues. I've seen this happen many times.
The tweet was also retweeted by OpenAI President Greg Brockman.
Netizens said this is a powerful application scenario:
Using GPT-5 Pro to verify scientific literature can greatly speed up the process of researchers verifying academic assertions and discovering logical contradictions.
There are also some netizens Amway tips:
The ultimate tip for poring over a scientific paper is to include in your prompts “please deep read - no grep, no scan - 1,000 lines at a time.”
Another suggestion is to do a circularity audit.
Erdös Problems official website: https://www.erdosproblems.com/faq
Reference Links:
[1]https://x.com/SebastienBubeck/status/1977181716457701775[2]https://x.com/gdb/status/1977153596811804890
This article comes from the WeChat public account "Quantum Bit" , author: Xifeng, and is authorized to be published by 36Kr.
