Terence Tao on Paul Erdős, the Erdős Number, and Discrepancy Theory

Terence Tao's Connection to Paul Erdős

• 💡 Terence Tao met the prolific mathematician Paul Erdős when he was 10 years old, during one of Erdős's visits to Adelaide.
• 🤝 Erdős was known for meeting bright young kids and treated Tao like an equal, later sending him a postcard with a math problem.
• 🧩 Tao recalls Erdős's famous quote: "Mathematicians are a means for turning coffee into theorems."

The Erdős Number Explained

• 📊 The Erdős number is a concept inspired by graph theory, reflecting collaborative distance to Paul Erdős in mathematical publications.
• 🔢 Erdős himself has a number of zero, direct collaborators have one, and collaborators of collaborators have two (Terence Tao's number).
• 🔗 Similar concepts exist, like the Kevin Bacon number for actors, and the Erdős Bacon number, which combines both and is usually infinite.

Erdős's Work Ethic

• ⚡ Erdős was known for his incredible productivity, with an anecdote suggesting he once bet to give up amphetamines for a month, only to claim it set mathematics back.
• ⚠️ This story highlights a past era with less emphasis on work-life balance compared to today.

Understanding Discrepancy Theory

• 🔬 Discrepancy theory examines the irregularity of sequences, such as those composed of plus ones and minus ones.
• 📈 A sequence can have low overall discrepancy (e.g., 500 plus ones, 500 minus ones in 1000 elements) but high discrepancy in subsequences (e.g., only even-indexed elements).
• ❓ Erdős posed the question of whether a sequence could be designed with bounded discrepancy over all homogeneous arithmetic progressions, meaning it would always remain balanced.

Solving the Erdős Discrepancy Theorem

• ✅ While some sequences can be extremely uniform for a while (e.g., 164 elements with discrepancy +/- 2), it was shown that eventually a discrepancy of three was required.
• ♾️ Terence Tao was able to prove that if these sequences are continued indefinitely, their discrepancy must eventually go to infinity, albeit extremely slowly (logarithmic or double logarithmic).
• 🛠️ This proof utilized advanced tools from information theory and number theory.