# What Is an Erdős Number?

*I am a former maths teacher and owner of Doingmaths. I love writing about maths, its applications and fun mathematical facts.*

## Who Was Paul Erdős?

Paul Erdős (pronounced "air-dush") was a Hungarian mathematician who worked predominantly in the fields of number theory and combinatorics.

Erdős was born in Budapest, Hungary, in 1913, the son of two high-school maths teachers. As a young child, Erdős spent a lot of time reading through his parents' maths books, developing a love for numbers early in life.

I fell in love with numbers at a young age. They were my friends. I could depend on them to always be there and always behave in the same way.

— Paul Erdős

Erdős showed great mathematical talent from a young age. At the age of three, he was multiplying three-digit numbers together in his head and by the age of four, he was working with negative numbers.

He attended Péter Pázmány University in Budapest from the age of 17 (despite his Jewish origins and the Hungarian restrictions against Jews in universities at the time), graduating with a PhD just four years later.

At this point, Erdős was effectively forced to leave Hungary due to antisemitism and took up a four-year post-doctoral fellowship in Manchester, UK. This was followed by moving to the USA in 1938 for a one-year stint at Princeton, New Jersey. At the end of this year, his fellowship was not renewed and Erdős began the travels for which he is often remembered today.

From the 1940s onwards, Erdős lived a nomadic lifestyle, turning down full-time job offers at prestigious universities in order to travel the world, often turning up unannounced on other mathematicians' doorsteps and staying with them while they collaborated on mathematics papers.

He continued to work on mathematics until his death from a heart attack in 1996. He was 83 years old.

Over the course of his life, Erdős published over 1500 papers and collaborated with 509 co-authors. It is this incredible number of collaborators that led to the creation of the 'Erdős Number'.

## The Erdős Number

As a tribute to Erdős and his incredible collection of published works, the Erdős number was created. This is calculated as follows:

- Erdős himself has an Erdős number of 0.
- Anybody who co-authored a paper with Erdős has an Erdős number of 1.
- Anybody who co-authored a paper with somebody with an Erdős number of 1 (but not directly with Erdős himself) has an Erdős number of 2.
- Each time the collaborator is one step further away, their Erdős number increases by 1.

A person who has never collaborated with anybody with an Erdős number has an Erdős number of infinity.

Over 40,000 people have an Erdős number of 3 or lower, and the current highest finite Erdős number is 15 (although this is almost certain to rise as mathematicians continue to work together in the future).

## The Erdős-Bacon Number

A fun extension of the idea of an Erdős number is the Erdős-Bacon number.

The Bacon number is calculated in much the same way as the Erdős number, although this time the link is starring in films with the US actor Kevin Bacon.

- Kevin Bacon himself has a Bacon number of 0.
- Anybody who has appeared in a film with Kevin Bacon has a Bacon number of 1.
- Anybody who has appeared in a film with somebody with a Bacon number of 1 (but not themselves appeared in a film with Kevin Bacon) has a Bacon number of 2, and so on.

For example, Colin Firth appeared in *Where the Truth Lies* with Kevin Bacon and so has a Bacon number of 1. Hugh Grant appeared in *Bridget Jones's Diary* with Colin Firth, but has not appeared in anything with Kevin Bacon and so has a Bacon number of 2.

Some people have both appeared in films and been credited on articles, hence have both an Erdős number and a Bacon number. The Erdős-Bacon number is created by adding these together.

Some examples of famous people with Erdős-Bacon numbers are:

Colin Firth: Erdős number = 6, Bacon number = 1, so Erdős-Bacon number = 7.

Stephen Hawking: Erdős number = 4, Bacon number = 2, so Erdős-Bacon number = 6.

**© 2020 David**