ChatGPT Erdős Problem Solved: Amateur Hits Record 1.6 Sum After 60 Years

Liam Price, a 23-year-old amateur mathematician, posted a record 1.6 Erdős sum for primitive sets on Erdős problem #464 on October 2, 2024. He used OpenAI's ChatGPT Pro tool. University of California, Los Angeles mathematician Terence Tao said the AI avoided common human errors in the proof process.

Price worked with undergraduate Kevin Barreto on the refinement. The result surpasses all previous records for the metric. Stanford University mathematician Jared Lichtman defined primitive sets as collections of positive integers where no element divides another element.

Primitive Sets Defined and Erdős Sum Calculation

Paul Erdős posed the problem in 1964. The Erdős sum equals the sum of the natural logarithms of the set's elements divided by the natural logarithm of the largest element. Erdős conjectured the sum approaches 1 as sets grow larger.

Price's achievement reached 1.6, exceeding prior benchmarks. Tao commented on October 8, 2024, in a Scientific American article: "Everyone who worked on the problem previously started by doing" standard proof steps.

Primitive sets relate to coding theory. Lichtman noted in his work that they aid in constructing sets with specific divisibility properties, which inform data compression in algorithms.

How ChatGPT Pro Generated the Proof

Price inputted the ChatGPT Erdős problem description into ChatGPT Pro, despite lacking prior knowledge. "I didn’t know what the problem was," Price told Scientific American. The AI produced an initial proof draft.

Price and Barreto iterated on the output multiple times. Lichtman reviewed it: "The raw output of ChatGPT’s proof was actually quite poor," he said. Powered by OpenAI's GPT-4 model, the tool allowed quick testing of multiple approaches.

This method reduced proof development from years to days, Lichtman added. AI tools enable non-experts to contribute to advanced research.

Terence Tao on the Key Human Error

Tao identified a shared oversight among experts. "Humans that looked at it just collectively made a slight wrong turn at move one," Tao said in the Scientific American piece. Researchers defaulted to conventional proof sequences.

Lichtman's 2022 Stanford doctoral thesis proved a related Erdős conjecture using traditional techniques. The AI approach skipped those paths and identified the optimal primitive set configuration.

Tao noted this demonstrates AI's strength in exploring unconventional mathematical strategies.

Applications in Blockchain Hashing and Cryptography

Primitive sets improve hash function design in blockchain systems. They enhance collision resistance by ensuring elements avoid divisibility relations that weaken cryptographic security, according to Lichtman's research on number-theoretic constructions.

Developers use such sets in consensus algorithms. Solana and Ethereum protocols incorporate optimized hashing derived from number theory principles. These improvements support faster transaction processing in decentralized finance (DeFi).

AI-assisted discoveries like Price's lower entry barriers for engineers building crypto infrastructure. Tao predicts large language models will tackle additional Erdős problems, advancing efficiency in algorithmic trading and smart contracts.

Crypto Market Snapshot on October 10, 2024

Bitcoin traded at $77,447 USD, down 0.3% over 24 hours with a market capitalization of $1,550.5 billion, according to CoinGecko data. Ethereum stood at $2,311.20 USD, also down 0.3% at $278.8 billion market cap.

• Asset: BTC · Price (USD): 77,447 · 24h Change: -0.3% · Market Cap (B USD): 1,550.5
• Asset: ETH · Price (USD): 2,311.20 · 24h Change: -0.3% · Market Cap (B USD): 278.8
• Asset: SOL · Price (USD): 86.05 · 24h Change: -0.4% · Market Cap (B USD): 49.5
• Asset: XRP · Price (USD): 1.42 · 24h Change: -1.2% · Market Cap (B USD): 87.6

AI-driven mathematical optimizations, including primitive set insights, enhance DeFi protocols. They boost transaction throughput and security amid rising market volumes. Price's work underscores technology-finance convergence in crypto ecosystems.