Chapter 1: A Unique Mathematical Mind

Paul Erdős stands out as one of the most significant and unconventional mathematicians of the 20th century. Throughout his career, he authored over 1,500 research papers, tackling a diverse array of mathematical topics. This remarkable volume of work is second only to that of Euclid. Notably, most of Erdős's publications focused on resolving existing mathematical challenges rather than proposing entirely new theories. His extensive collaboration with fellow mathematicians was key to his prolific output.

Erdős is also famous for his peculiar way of living. He did not maintain a fixed residence; instead, he roamed between the homes of friends and various conferences, relying on their hospitality for accommodation and meals, often expecting them to do his laundry too! Those who accommodated him frequently found themselves benefiting from his intense focus on collaboration, leading to numerous novel mathematical results. Erdős partnered with over 500 people, giving rise to the concept of the "Erdős number," a playful metric indicating one's collaborative distance from him. Erdős himself holds the number 0, while his coauthors are numbered 1, and those who coauthor with them are assigned the number 2, and so forth.

Erdős championed the notion that presenting mathematical problems was as crucial as solving them. He often published new challenges, offering financial rewards for their solutions. One notable challenge he posed was the Collatz Conjecture, for which he offered a $500 prize to anyone who could resolve it. Erdős had a particular appreciation for simple and elegant proofs, frequently alluding to "The Book," which he described as:

"A visualization of a book in which God had written down the best and most elegant proofs for mathematical theorems." — Paul Erdős

Mathematicians have since compiled a collection of proofs deemed elegant enough to belong in "The Book," with significant contributions from Erdős himself. Some mathematical theorems boast multiple elegant proofs; for instance, there are six distinct proofs for the assertion that there is an infinite number of prime numbers, ranging from elementary to highly sophisticated.

Given his penchant for simplicity, Erdős diligently sought the most elegant resolution to each problem. This was exemplified by his novel proof of Bertrand's Postulate, which had already been established decades earlier. Erdős discovered a much more streamlined version when he was only 19, garnering considerable attention.

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This video explores Erdős's mathematical journey and his impact on modern mathematics.

Section 1.1: Bertrand's Postulate

Bertrand's Postulate is straightforward to grasp. If you select a number ( n ) greater than 1, there is always a prime number between ( n ) and ( 2n ). Bertrand himself demonstrated its validity for numbers up to 3,000,000 by hand and suspected it held true for all ( n ) above 1. We can easily verify this for a few small numbers.

However, how can we prove this assertion for all ( n )? Erdős approached this question using principles from combinatorics, relying on various results to construct his elegant proof. For a detailed examination of his work, a link is provided at the end of this article.

Subsection 1.1.1: The Happy Ending Problem

The Happy Ending Problem presents another accessible theorem with a charming backstory. Given any set of five points, it is possible to connect four of them to create a convex quadrilateral. "Convex" means that none of its angles bend inward. In the accompanying image, I illustrate three examples of five-point sets and the convex quadrilaterals that can be formed from them. Esther Klein collaborated with George Szekeres and Paul Erdős in the 1930s to resolve this problem.

The trio proved the theorem by categorizing all potential arrangements of five points into three distinct cases, assuming that no two points occupy the same position and that no three points lie on a straight line, a scenario known as "general position."

The original idea came from Klein, who demonstrated that the five points could form a convex pentagon, quadrilateral, or triangle with the remaining points situated in the middle. The images above show how to transition from each of these three scenarios to create a convex quadrilateral. While there are additional mathematical intricacies involved, this encapsulates the fundamental concept of their proof.

The theorem is named after Paul Erdős and the context surrounding it. Although Klein and Szekeres were acquainted before their collaboration, their work on this problem brought them closer together, eventually leading to marriage, hence the title "Happy Ending Problem." Erdős believed that resolving this problem played a pivotal role in their burgeoning relationship.

Considered a lighthearted mathematical challenge, this theorem laid the groundwork for a critical area of mathematics known as Ramsey Theory. This field focuses on establishing what types of ordered structures must emerge from random arrangements. For example, this theorem confirms that a certain shape must be present in any arbitrary configuration of five points.

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This video provides insights into Erdős's significant contributions to mathematical theory.

Going Further

I hope you discovered something enlightening! Paul Erdős was an extraordinary mathematician whose extensive contributions span all areas of mathematics. This article only scratches the surface of his varied work, sharing merely two of his notable contributions. For those interested in delving deeper, I've included a few links below for further reading.

• An engaging biography titled The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth offers a captivating blend of his life story and mathematical endeavors.
• A maintained version of "The Book" that Erdős frequently referred to is linked here. While some of the mathematics may be complex, it is undoubtedly rewarding!
• A comprehensive list of Erdős's problems, both solved and unsolved, is available here.
• Another site keeps a complete record of Erdős numbers—it's fascinating to explore!
• This excellent page provides an overview of Erdős's proof of Bertrand's Postulate.

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