If you want to explore Egyptian Fractions more, here is a paper focusing on the Erdős-Graham problem [1.]: "On a Coloring Conjecture about Unit Fractions" by Ernest Croot [2.]
He was the most prolific mathematician of all time (most published papers) by an order of magnitude. His papers launched entire subfields of mathematical inquiry. He dedicated his life to math in a way that most people could never dream of dedicating themselves to anything, having no home or family or possessions and instead doing mathematics all day every day. These seem like good reasons for him to be popular among mathematicians.
An order of magnitude is usually 10. But Erdos only published about twice as many papers as Euler (if this is even a good way to compare their respective outputs).
Also, Erdos was not so much of a theory builder but a problem solver. He also left many unsolved problems, some of which have led to new insights. But he didn't spark the development of new fields of maths in the way that say Grothendieck did.
For anyone interested in learning more about this cultural divide within mathematics -- theory building versus problem solving -- Fields medalist Tim Gowers has an interesting article: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf.
People I talk to are often surprised to learn that there is quite a bit of snobbery within mathematics on this issue. Mostly people in theory-heavy subjects looking down on areas that revolve more around problem solving. Of course, it's academia, so it surprises me that anyone is surprised that there are turf wars and snobbery.
"Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time. I happen to be a frog, but many of my best friends are birds."
I was not aware that Euler had so many publications. Good point.
However, even though I believe Erdos's insights were as deep as Grothendieck's, and that Erdos was as great a mathematician, I don't think I could ever convince an algebraic geometer of it.
"The Man Who Loved Only Numbers: Story of Paul Erdos and the Search for Mathematical Truth"
Is a humorous non-mathematical book about him. Erdos was a funny and strange man. He was the most prolific co-writer of math papers.
He didn't have a home, but would just pack his bags and sleep over at other mathematicians couches around the world. He would ask about their math interests and problems. Erdos often had some great insights and unusual perspectives about it. He would talk about math with them non stop, until they could take no more. The other mathematician would often write a paper about what they had discussed, and Erdos would be sent on his way to the next mathematician.
When he discovered or heard about some very beautiful proof. He would say this proof was "from The book". As in the book that God wrote. He also called God the supreme fascist.
Erdos dint sleep much and didnt care about things that where not math. He couldn't tie his shoes, liked to tell silly jokes, and was a virgin. He was also on amphetamines most of the time.
You likely mean something in the vein of The Theory of Everything or The Imitation Game rather than a straight documentary but just in case documentaries count: it's called N is a Number.
Incidentally, through his appearance in N is a Number, Erdös has a Bacon number of 3.
> After 1971 he also took amphetamines, despite the concern of his friends, one of whom (Ron Graham) bet him $500 that he could not stop taking the drug for a month.[18] Erdős won the bet, but complained that during his abstinence, mathematics had been set back by a month: "Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper." After he won the bet, he promptly resumed his amphetamine use.
I think part of it is that Erdos worked with an unusually large number of people, at an unusually large number of places. Another reason is that he was quite eccentric, even by the standard of mathematicians. There are so many stories about him, which led to a sort of legend, and that led to all the books and documentaries. Finally he was around at a time when discrete math was gaining traction as a de facto area of mathematics, and he made a great number of contributions in that area. For example the probabilistic method he popularized is a cornerstone of the subject nowadays.
Whether or not he was disproportionately popular depends on what you feel it should be proportionate to. I reckon he is more famous than similarly important figures in other subfields of math. But few mathematicians have worked with and influenced so many people.
obviously this is not an actual new erdos paper, but without these shenanigans everyone's erdos number would slowly but surely increase, as first nobody new could have co-published with him (too young) and soon the same applies to his co-authors.
while it might have been cute back in that era, today I think it should be treated as a historical thing.
I'm not sure what you mean by "shenanigans". Erdős was legitimately a collaborator. Going by the article, Graham and Erdős worked together to try to tackle a similar problem but never managed to make it work. Then Graham and Butler built on that to arrive at this. It doesn't seem relevant that Erdős' contribution happened decades ago.
Since he's still coauthoring papers, I prefer to think that Erdős has merely left.