The doorbell rings, and you’re surprised to find your colleague on the stoop. He’s slight, elderly, buzzing from amphetamines, unkempt and uninvited. He shoulders past you into your living room, a single suitcase containing all of his worldly possessions in tow, and declares, “My brain is open.” You have no idea how long he intends to stay because he doesn’t have a house of his own to return to. You’re expected to do his laundry and cook his meals because he can’t be bothered to learn to take care of himself. In exchange, you’ll receive a sleepless whirlwind communing with one of the greatest mathematical minds of the 20th century. Your involuntary hospitality will likely result in an academic publication with your name on it.

This was many people’s experience of Paul Erdős, the most prolific mathematician of all time.

Erdős (pronounced *air*-duhsh—the ő has a Hungarian double acute accent mark, not an umlaut) was born in Budapest in 1913 to two high school math teachers. He was a pampered prodigy. By age four he could calculate in his head how many seconds a person had been alive, and at age 21 he buttered his own bread for the first time. That same year he earned his Ph.D. in math. His subsequent fellowship position at Princeton University was cut short because, according to *The Man Who Loved Only Numbers,* Paul Hoffman’s biography of Erdős, “they found him uncouth and unconventional.” Thus began his nomadic life, in which he flitted between brief academic stints, conferences and friends’ guest rooms. As he would say, “Another roof, another proof.”

Erdős was a notoriously bad houseguest. In Hoffman’s book, mathematician Michael Jacobson recounted a story in which Erdős came to his home, and they worked on math until 1 A.M., when Jacobson finally succumbed to exhaustion. Erdős, who tended to put in 19-hour days, stayed up and, at 4:30 A.M., banged pots in the kitchen incessantly to wake up his host. Jacobson eventually teetered downstairs in his bathrobe and recalled the ensuing interaction with his colleague: “What were the first words out of his mouth? Not ‘Good morning’ or ‘How’d you sleep?’ but ‘Let *n* be an integer.’”

Erdős’s single-minded obsession with math led to his authorship of a whopping 1,500-plus academic publications, more than any other mathematician in history. Quick aside: some contend that 18th-century mathematician Leonhard Euler was the most prolific of all time. Indeed, Euler produced more *pages* of math, while Erdős produced more papers. So who holds the crown depends on the unit of measure, but the top two most prolific mathematicians are uncontroversial.

One of Erdős’s most notable contributions was to something called the probabilistic method. To illustrate its value, imagine that you’re planning a mixer for 100 people, and because mixers work best when some partygoers already know each other and some don’t, you want to guarantee that no group of six guests are all friends or all strangers. Is that even possible? If you try to avoid groups of strangers by inviting many friends, then it becomes harder to avoid cliques of friends,. yet too many strangers will yield the opposite problem.

Mathematicians often want to prove the existence of a mathematical object with certain properties, such as our desired party of 100 people. A natural way to prove its existence would be to give an explicit example of such an object (for instance, come up with a guest list with no groups of six mutual friends or strangers). This may be quite difficult in practice, however.

Instead Erdős pioneered an ingenious alternative. Rather than trying to design the guest list by hand, just pick that list (or whatever type of object you’re trying to find) completely at random. Then change your question to: What is the probability that my randomly chosen object has my desired properties? If you can prove that the probability is anything greater than zero, then voilà! Your object must exist; if it didn’t, the probability would be zero.

Changing the question to one about probability often makes it easier to answer. That’s in part because it now allows you to apply a rich set of tools from probability theory. Interestingly, because the probabilistic method circumvents the need to construct your object, you often end up knowing that something exists but having no clue what it looks like. Erdős cracked many stubborn math problems with the probabilistic method, including a more general version of our mixer problem. Today the method is considered an essential technique in every researcher’s tool kit.

Much of his success sprung from his belief in math as a social activity. He had so many collaborators that the field invented the Erdős number, a measure of authorship distance from Paul Erdős, which serves as a badge of honor for scholars. Everyone with whom Erdős co-authored a paper has an Erdős number of one, while all of *their* co-authors have a two, and so on. You may have heard of the Bacon number, an actor’s co-starring distance from Kevin Bacon, but Erdős’s recognition as the center of his network predates Bacon’s by 25 years.

Researchers have devoted a surprising amount of effort to investigating the Erdős number, both as a lightweight amusement and as a serious tool for understanding connectivity patterns in authorship networks. Here are some curious facts about it:

- Of the more than a quarter-million mathematicians who share an authorship chain with Erdős, the median number of hops required to reach him is five. (I’m proud to have an Erdős number of three.)
- Many prominent figures beyond math have Erdős numbers: e.g., Noam Chomsky (four), Angela Merkel (five), Stephen Hawking (four) and Elon Musk (four).
- If one is in a playful mood, they might argue that Baseball Hall of Famer Hank Aaron has an Erdős number of one because the two men signed the same baseball when they received honorary degrees together from Emory University.
- Actress Natalie Portman boasts the rare distinction of having an Erdős number (five)
*and*a Bacon number (two) because of her neuroscience publication as an undergraduate. (Natalie Hershlag is her birth name.) - Someone once tried to sell an Erdős number on eBay. The winner would get to collaborate with the seller, whose Erdős number was four. Several people placed substantial bids, but the auction was snagged at the last second for $1,031 by a mathematician with no intention to pay up, who called the stunt a “mockery” of the system.

Erdős’s legacy lives on not just through his publications but in the many conjectures he left behind. Sometimes the hardest thing in math is asking the right questions, and he had a keen talent for pinpointing important problems. He issued personal monetary prizes for many problems despite having little of his own money. What he collected through speaking fees, awards and part-time appointments, he typically donated to unhoused people, charities and aspiring researchers. He once gave $1,000 to a talented high school student struggling to meet tuition for Harvard. Ten years later that student felt ready to pay back Erdős, who instead insisted, “Do with the $1,000 what I did.”

Paul Erdős was a man devoted to exactly one thing. He never married or had children—in fact, he was celibate his entire life. He had very few hobbies, didn’t drive and didn’t have a permanent residence or a permanent job. Erdős died in 1996 at a math conference in Warsaw. He died doing what he loved, largely because he never did anything else.