Paul Erdős: How I became a mathematician and a world traveller

The Ludwig Múzeum in Budapest and the Neue Galerie in Graz organized an exhibition called Beyond Art (A mqűvészeten túl) for the Austrian millennium and 1100th anniversary of Hungary. The exhibition tried to illustrate, with well-chosen examples, the joint cultural and spiritual history of Austria and Hungary. Introducing some of the outstanding individuals from our mathematics also had an emphasis. The organizers of the exhibition asked Paul Erdős in the summer of 1996 to write about himself and how he became a mathematician. As the art historian ladies asked, Uncle Paul took out his pen and started writing about himself in his ruled notebook. Sadly, for the last time ...

We publish this writing below, in a translation by University of St Andrews undergraduate Kamilla Rekvenyi.

I was born on 26 March 1913, my parents were mathematics teachers, they met at university. During my birth my parents were hit by a terrible tragedy. I had two sisters, 3 and 5 years old, and while Mother was in the hospital, during my birth, within 24 hours, both of my sisters died from scarlet fever. That time of course there was no penicillin. Mother brought it up several times later how lucky it was that she had to teach and pull herself together, but she could never forget this terrible tragedy.

In the beginning of 1914 my father was called to the military. He was taken to the Russian front and became a prisoner of war shortly after in August 1914. He spent 6 years in Siberia. Being a military officer, he was relatively well looked after and he learnt English and French in captivity. Mother was teaching so I spent a lot of time alone with an Austrian lady, so I learnt German at the age of 3. My parents also spoke good German. Mother was born in Vágbeszterce. Her first languages were Slovak and German, but she spoke perfect Hungarian. My father was born in HódmezQvásárhely.

When I was around 3 or 4 years old I could already count well, since I wanted to know when the vacation comes, so Mother ddidn't need to go to school. I often played with the calendar and my mental arithmetic developed early on. My biggest discovery was, that I told Mother at the age of 4: if we subtract 250 from 100 we get 150 under 0, when of course Mother told me about the existence of negative numbers. I didn't go to elementary school, I learnt with Mother, and when I took my exams from the first year of school, they immediately examined me on the second year curriculum, so I gained a year.

In 1920 my father returned home in a French ship via Vladivostok and we started learning English. When I was 10 he told me the proof that there are infinitely many primes and that there are arbitrarily big gaps between primes, so my friendship with primes started very early. I learnt a lot of mathematics from my parents; higher pure mathematics mainly from my father. At the age of ten I was still a Hungarian Nationalist, but my father argued a lot with me about it, so at 12 I became an Internationalist. Unfortunately half of Hungary was fascist and anti-Semite, so I knew I would have to go to Western Europe whenever it would be possible. I started focusing on mathematics very early on, but I wasn't such an outstanding child prodigy as my later students, Pósa or Lovász.

I want to tell another early story, which might explain my independence. In the end of 1919 and 1920 it was common to see Jews beaten up on the street (especially when they were suspected of being communists). Mother once asked whether we should convert to Christianity. I replied:  You can do whatever you want, but I'll stay what I was born to be. Being Jewish, as a matter of fact didn't mean anything to me, but I didn't like, probably instinctively, when outer forces ordered me around. This might explain my saying from later on: neither Sam, nor Josef can determine where and when I can travel. [Sam = Uncle Sam = USA, Josef = Stalin = Soviet Union].

Enough about me, there are more details in the papers of Babai and by Bollobás in a book in honour of my 80th birthday.

He started writing in his notebook ... and thought about problems

Now I'll turn to mathematics. I would like to begin with prime numbers. In my first paper I found a new proof for Chebyshev's Theorem, that there is a prime number between n and 2n for all n. Let pn be the nth prime, I proved in the end of 1933 that for infinitely many n

pn+1 - pn > c log n log log n/(log log log n)2
In 1938 Rankin slightly sharpened the theorem. He added an extra log log log log n factor to the numerator. This hasn't been further updated significantly. So nobody has proved, that there exists some f (n), such that f (n) tends to infinity as n tends to infinity and for infinitely many n
pn+1 - pn > f (n) log n log log n log log log log n /(log log log n)2
I would give at least $3000 for a proof of the above. Presumably the maximum of pn+1 - pn is of the order of (log n)2, but this question might not be answered for the next few thousand years.

Number theory is full of incredibly simple, and seemingly almost unsolvable problems, Edmund Landau listed four of these in the international conference in Cambridge in 1912, whose solutions are "unreachable with our current scientific knowledge". These are the as yet unsolved problems:

  1. Every even number is a sum of two primes [Goldbach's conjecture]
  2. There are infinitely many twin primes, so there are infinitely many primes pn such that pn + 2 is also a prime [Twin primes conjecture]
  3. There is a prime between any two squares [Lagrange's conjecture]
  4. There are infinitely primes of the form n2 + 1
We already know this much: that every significantly large number is the sum of three primes and for significantly large n there exist a prime between n3 and (n+1)3.

Maybe it's enough for me to tell my first serious problems from 1931. There was a $500 prize promised for the solution.

Let 1 ≤ a1 < a2 < ... < akn and suppose that the sums of all the subsets of these integers are different. This is a property of, for example, the powers of 2.
Is it true that max k < (log n/log 2) +c, where c is a constant that doesn't depend on n? Maybe k is a maximum r + 2, if n = 2r?
I have a lot more results in number theory. Maybe one of my most significant result is my collaboration with Kac, which was probably one of starting points of the application of probability theory in number theory, and which hopefully exceeds its authors with centuries. Our work started in March of 1935, and we wouldn't have been able to complete it individually, since I didn't know enough probability theory and Kac didn't know enough number theory. It was a good example of the usefulness of collaboration.

Due to the shortness of this article I won't mention my paper with Turán about interpolation and statistical group theory, our results with Hajnal and others in set theory and with Sárközy and Szemerédy in number theory. I will also omit my papers in number theory and graph theory with Vera T Sós ( Paul Turán's wife and co-worker). I only want to mention another theorem from number theory and geometry. Selfridge and I proved that the product of two consecutive numbers can't be a power.

In 1932 Eszter Klein (later wife of Szekeres) asked:

What is be the smallest number f (n) such that any f (n) points on the plane (with no three collinear) always determine a convex n-gon?
She proved that f (4) = 5. An earlier result by Makai and Turán was f (5) = 9. Szekeres and I proved that
2n - 2 < f (n) < 2n-4Cn-2
Probably f (n) = 2n-2 + 1, but f (6) = 17 has not been proved.

Remarks: Erdős named this the happy ending problem since the two early workers, Klein and Szekeres became engaged and eventually married. A computer search (Szekeres and Peters, 2006) proved that indeed f (6) = 17.

I have probably written more papers than anyone else, and I have collaborated with the highest number of people. In the old Hungarian Parliament, where not only nobles appeared, they used to say, the votes shouldn't be counted, but evaluated. This wasn't correct and democratic in politics, but it surely is in science.

JOC/EFR January 2017

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