Paul Erdős was easily the most influential mathematician of the 20th century, and arguably so for all time. He published 1475 papers almost all in collaboration. Mathematicians have Erdős Numbers. Your number is 1 if you co-authored with him, 2 if you co-authored with a co-author, and so on. Movie star Natalie Portman has an Erdős number of 7. Danika McKellar’s 4 is lower than her Kevin Bacon number.
Erdős’s work was beyond prolific. He knew how to offer motivating challenges to people working at all levels of mathematics from his academic peers to children. In that, Paul Erdős was responsible for hundreds of proven insights that extended the frontiers of number theory.
The fact that Erdős’s life (1913-1996) intersected so many others allowed Paul Hoffman’s biography to explore the domain and range of the history of mathematics. The Greeks, Fibonacci, and pi are here along with Hardy, Ramanujan, and transfinite numbers, as well a bit of graph theory, and “what’s behind door number two?”
The Man Who Loved Only Numbers by Paul Hoffman (Hyperion, 1998) explains most of the mathematics with integers. After all, God created the integers and we built the rest – or so it was claimed by Leopold Kronecker (1821-1893) and echoed by Stephen Hawking. As a result, many of these puzzles could be explained to a child in third through ninth grade. The fact is, though, that few would be. The stampede for standardized testing in K-12 education forces teachers to focus on the examinations to the detriment of the true understanding that comes from the artful competence of leisure and play.
Consider Ramsey theory. Among the pursuits of Frank Plumpton Ramsey (1903-1930) was the question of the smallest possible "universe" that contains some element. How many ordinary people would you have to fetch at random in order to be guaranteed one of each sex (not gender)? Three, right? If you want to plan a party, what is the smallest number of guests that guarantees that three of them must know each other? Six; no proof is offered, so it must be hard. (The opposite problem that no three of them will know each other is the same problem, again stated without proof.) If you wrote out the first so-many integers in any order you wanted, how many would you need to guarantee a run of eleven in a row ascending (or descending)? 101, but 100 might work for special cases. To find a string of length n+1, you must have a universe of n^2 + 1. Anyone in a first-term computer programming class could write a "Ramsey generator."
Reading the book while commuting to work on the city busses, I misread one of the problems and worked a different one entirely. It so happens that any odd positive integer raised to any integer power can always be expressed as the sum of two consecutive integers: 9^2 = 81; 81= 40+41. 7^5 = 16807 = 8403+8404.
Suggestive as isolated cases may be, in mathematics, we need proof, and the more general the proof, better. Best of all is a simple proof. And a proof must reveal not merely that something is true, but why it must be true. As abstract as mathematics is, when you work with integers these necessary truths became necessary factual truths because party guests and anything else we count are sensible evidentiary empirical objects.
So, one morning, I started with 2n+1, the common form of an odd number. (2n+1)^2 = 4n^2 + 4n + 1. That can be written as (2n^2 + 2n) + (2n^2 + 2n + 1), clearly some number and the next higher. The next night, I did the same for cubes. The following day, I had to open up a math book to see how to write out the expansion for any power n, an algebraic statement for Pascal’s Triangle. I was pretty sure that I could complete the proof. Then I realized that if an odd number can be expressed as 2n+1, that 2n is always some integer that admits to the existence of n and the next number would be one more than that. More to the point, no matter what power (2n+1) is raised to, the last term of the polynomial will be 1. You always will be able to find half the number and the integer next to it. (I called it "Proving Gershon's Theorem" after the Sidney Harris cartoon: "You can't call it Gershon's Equation if everyone has known it for centuries.")
Then I tried it with negative integers. They only work with odd powers. And I can prove why. It is child’s play, really. But few people ever approach mathematics that way. Paul Erdős did. In fact, he exhibited neoteny, never having any intimate relationships, being cared for by his mother into his sixties, being unwilling to cook for himself or otherwise look after the simplest daily tasks. Instead, he was in constant motion, traveling to visit colleagues, imposing on their hospitality, in return for which, he gave them the impetus to publish over 1400 significant new ideas in mathematics.
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