Paul Erdos (1913—1996)

Paul Erdős was born in Budapest, Austria-Hungary, on 26 March 1913 Paul Erdős was born in Budapest, Austria-Hungary, on 26 March 1913. He was the only surviving child of Anna and Lajos His two sisters, aged 3 and 5, both died of scarlet fever a few days before he was born. His parents were both Jewish mathematics teachers from a vibrant intellectual community. Paul was a year old when World War I broke out. Paul's father Lajos was captured by the Russian army as it attacked the Austro-Hungarian troops. He spent six years in captivity in Siberia. As soon as Lajos was captured, with Paul's mother Anna teaching during the day, a German governess was employed to look after Paul. Anna, excessively protective after the loss of her two daughters, kept Paul away from school for much of his early years and a tutor was provided to teach him at home. Erdős received much of his early education from his mother and, after the was ended, from his father. His fascination with mathematics developed early. By the age of four, given a person’s age, he could calculate, in his head, how many seconds they had lived.

In 1920 Paul’s father returned home from Siberia In 1920 Paul’s father returned home from Siberia. He had learnt English to pass the long hours in captivity but, having no English teacher, he did not know how to pronounce the words. He set about teaching Paul to speak English … with a strange accent that remained one of Paul’s characteristics throughout his life. https://www.youtube.com/watc h?v=EGc6rE24YSw

Despite the restrictions on Jews entering universities in Hungary, Erdős, as the winner of a national examination, was allowed to enter in 1930. As a college freshman, he made a name for himself in mathematical circles with a stunningly simple proof of Chebyshev’s theorem, which says that a prime can always be found between any integer (greater than 1) and its double n < p< 2n

Erdos studied for his doctorate at the University Pázmány Péter in Budapest. He was awarded a doctorate in 1934, (Age 21!). Then he took up a post-doctoral fellowship at Manchester, essentially being forced to leave Hungary because he was Jewish. During his tenure of the fellowship, Erdős travelled widely in the UK. He met Hardy  and Ulam among others. The situation in Hungary by the late 1930s clearly made it impossible for someone of Jewish origins to return. In March 1938 Hitler took control of Austria and Erdos had to cancel his intended spring visit to Budapest. Within weeks Erdos was on his way to the USA where he took up a fellowship at Princeton. He hoped for his fellowship to be renewed but he did not conform to Princeton's standards so he was offered only a six month extension rather than the expected year.

The truth is that Princeton found Erdos … … uncouth and unconventional The truth is that Princeton found Erdos … … uncouth and unconventional... And that’s why his fellowship was not renewed

In later years , Princeton relaxed its definition of “conventional” and focused more on “genius” …

(By the way, the guy in the second picture of the previous slide is John Nash, “A beautiful mind”. John Nash died in 2015in a car accident

Erdos began his “Brownian motion” from university to university, always with visiting short-term positions. He would not stay long in one place and traveled back and forth among mathematical institutions in 22 countries until his death.

A rather silly event that took place in August 1941 was to have any real effect on Erdős's life: Erdős and two fellow mathematicians were picked up by the police near a military radio transmitter on Long Island. It was just three mathematicians who were being too absorbed in discussion of mathematics to notice a NO TRESPASSING sign. After a friendly session with the police it was realized that no harm had been intended. However, this incident gave Erdős an FBI record which was later used against him.

During the early 1950s senator Joseph R McCarthy whipped up strong feelings against communism in the United States. Erdős began to come under suspicion from authorities. By then Hungary was a communist country in the block of the Former Soviet Union. When asked by US immigration, as he returned after a conference in Amsterdam in 1954, what he thought of K. Marx, Erdős made the ill judged reply:- “I'm not competent to judge, but no doubt he was a great man” This was followed by a line of questioning about whether he would ever return to Hungary. Erdős said:- I'm not planning to visit Hungary now because I don't know whether they would let me back out.” So, it was only the fear of not being let out of Hungary that stopped him going there. When questioned further about whether he was planning to go back to Hungary in a near future Erdos replied innocently:- Of course. My mother is there and I have many friends there.

Erdős was not allowed back to the United States ; the interview with the US immigration, the fact that he had corresponded with a Chinese mathematician who had subsequently returned from the United States to China and also his infamous 1941 FBI record made him “persona non grata” …

Erdos spent much of the next ten years in Israel Erdos spent much of the next ten years in Israel. During the early 1960s he made numerous requests to be allowed to return to the United States and a visa was finally granted in November 1963. By this time, Erdős had become a traveler moving from one university to another, and from the home of one mathematician to another (where he often arrived unannounced). He did have a home of sorts with his friend Ronald Graham. Erdős and Graham met at a number theory conference in 1963 and soon began a mathematical collaboration. It was Graham who provided a room in his house where Erdős could live when he wanted; he also stored Erdős's papers there and, in many ways, acted as a secretary to Erdős.

With amphetamines to keep him going, Erdős did mathematics with a missionary zeal, often 20 hours a day, turning out some 1,500 papers, an order of magnitude higher than his most prolific colleagues produced. His enthusiasm was infectious. He turned mathematics into a social activity, encouraging his most hermetic colleagues to work together. Erdős himself published papers with 507 coauthors.

The Erdos number My Erdos number is 4! In the mathematics community those 507 people gained the coveted distinction of having an “Erdős number of 1,” meaning that they wrote a paper with Erdős himself. Someone who published a paper with one of Erdős’s coauthors was said to have an Erdős number of 2 … and so on. Non-mathematician have an Erdos number equal to infinity My Erdos number is 4!

The prime number theorem In 1948 Erdős and Atle Selberg proved the celebrated prime number theorem  Let π(x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that x / log(x) is a good approximation to π(x), in the sense that the limit of the quotient of the two functions π(x) and x / log(x) as x goes to infinity, is 1. The prime number theorem was first was conjectured independently by Legendre and Gauss at the end of the 18th century. Partial results were proved by Tchebychef Sylvester and others (~ 1850).

The Erdos-Selberg dispute arose over the question of whether a joint paper on the entire proof or separate papers on each individual contribution should appear on the elementary proof of the Prime number theorem. Erdos wrote the paper “On a new method in elementary number theory which leads to an elementary proof of the prime number theorem,” (Proceedings of the National Academy of Sciences , 1948). At the same time, Selberg rushed ahead and published the whole result “An elementary proof of the prime–number theorem,” in the Annals of Mathematics. Even though he credited the contribution of Erdos, the paper is authored only by him. In 1950 Selberg was awarded the Field medal for his work; In 1951 Erdos won the (far less prestigious and lucrative) Cole Prize

In 1984 Erdos won the most lucrative award in mathematics, the Wolf Prize, and used all but $720 of the $50,000 prize money to establish a scholarship in his parents’ memory in Israel. He was elected to many of the world’s most prestigious scientific societies, including the Hungarian Academy of Science (1956), the U.S. National Academy of Sciences (1979), and the British Royal Society (1989). Defying the conventional wisdom that mathematics is a young man’s game, Erdős went on proving and conjecturing until the age of 83. He died of a hart attack in 1996, at a conference in Warsaw.

Link to the New York times obituary: http://www-history.mcs.st- andrews.ac.uk/history/Obits2/Erd os_NYTimes.html Erdos proposed many problems and often offered money for their solution. A complete list of Erdos’ conjectures can be found e.g. on Wikipedia.

Conjecture on powerful numbers A powerful number is a positive integer m such that for every prime number p dividing m, we have that p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Unsolved problem Can three consecutive numbers be powerful?

Arithmetic progressions Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic . It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions. Because the sum of the reciprocals of the primes diverges, the Green– Tao theorem on arithmetic progressions, proved in 2004, is a special case of the conjecture Formally, the conjecture states that if Sum 1/x= infinity , with x in A Then A contains arithmetic progressions, i.e. elements in the form of x= p+ n q, with n= 1, 2, … N of any given length.

The Egyptian fraction problem posed by Erdos is solved! https://www.simonsfoundation. org/uncategorized/new-erdos- paper-solves-egyptian-fraction- problem/

The Erdos number theory project http://wwwp.oakland.edu/enp/

Erdős's vocabulary was very peculiar: Children were referred to as "epsilons" (because in mathematics, particularly calculus, an arbitrarily small positive quantity is commonly denoted by the Greek letter (ε)) Women were "bosses". Men were "slaves". People who stopped doing mathematics had "died". People who physically died had "left". Alcoholic drinks were "poison". Music (except classical music) was "noise". People who had married were "captured". People who had divorced were "liberated". To give a mathematical lecture was "to preach". To give an oral exam to a student was "to torture" him/her. He gave nicknames to many countries, examples being: the U.S. was "samland" (after Uncle Sam), the Soviet Union was "joedom" (after Joseph Stalin), and Israel was "isreal".