[Note: The story below was excerpted from an article that appeared in the Chicago Tribune, “Erdös left a whole world of survivors”, written by Charles Krauthammer in the Fall of 1996. For a more lengthy account of his life, click here. For information about an interesting book about his career, click here.]
Paul Erdös was his name; and math was his game. His only game, in fact. Unlike most of us, mathematician or not, “he had no home, no family, no possessions, no address…. [He] traveled with two suitcases, each half-full. One had a few clothes, the other, mathematical papers. He owned nothing else. Nothing. His friends took care of the affairs of everyday life for him—checkbook, tax returns, food. He did numbers.”
Born in Hungary in 1913, he lost his two sisters from scarlet fever. Later much of the rest of his Jewish family was lost to the Hitler regime in the ‘30s. Erdös himself never married, hence when he passed away, the newspaper obituary ended with these sad words: “He leaves no immediate survivors.”
But as the columnist points out, that really wasn’t completely true. Erdös had many “survivors” in the many people whose lives he influenced with his work. “He went from math conference to math conference, from university to university, knocking on the doors of mathematicians throughout the world, declaring ‘My brain is open’ and moving in.” He worked with many mathematicians, collaborating with them to produce hundreds of math articles. He gave generously of his vast knowledge to help others with difficult topics.
One of his friends tells the following story about how Erdös helped others in another, more human interest way. It seems Erdös “heard of a promising young mathematician who wanted to go to Harvard but was short of the money needed. Erdös arranged to see him and lent him $1000…[saying] he could pay him back when he was able to.” (Know that Erdös seldom had more than $30 in his pocket at any one time.) Later, when the young man graduated and was working, he asked Erdös’ friend how he should repay the loan. Erdös told the friend to inform the man, “Tell him to do with the $1000 what I did.”
An ERDÖS Problem
One of the classic math problems that Erdös is famous for providing a proof concerns prime numbers (a favorite topic here at WTM). At the age of 20, Erdös showed once and for all that between any number (i.e. integer greater than 1) and its double there must exist at least one prime. For example, between 10 and 20 there are four primes (11, 13, 17, 19). It’s easy to see that it’s true for many examples, but it’s another thing to prove it’s true for all integers. And that’s what Erdös did.WTM would like to play with this idea a little bit by inventing a new idea, called the Erdös Value of a number. It is simply the number of primes between a given number n and its double 2n. We will write things as follows, using the example just cited:
EV(10) = 4,because there are four primes between 10 and its double, 20.
Now here is a WTM problem for you to think about: what is the smallest number n whose Erdös Value is 10?
Or put in terms Mr. Erdös would like:
Find the least positive integer n such thatEV(n) = 10.
The notation used above to write the Erdös Value shows that we have a function on our hands. But it’s not your usual, everyday function that you meet in your algebra courses. While it is a generally increasing function — which means, as n gets bigger, f(n) gets bigger also — it isn’t a strictly increasing function. Make a line graph of EV(n) over the set of numbers from 2 to 50 to see what I mean here. It’s a new way to think about what “function” really means. For an example of another new function, U(n), that I invented, see Ulam.