Paul Erdos

Paul Erdos was born in Budapest on March 26, 1913, into a Jewish family of mathematics teachers. His parents, Lajos and Anna, both taught the subject in secondary schools, and the household treated numbers as a natural language. The family's story opened with tragedy: his two older sisters died of scarlet fever within days of his birth, and his mother, fearful of losing another child, kept him at home for much of his early schooling ^[1].

The boy's talent surfaced startlingly early. Family accounts describe him multiplying three-digit numbers in his head at age three and independently arriving at the idea of negative numbers around age four ^[1]. His father, held as a prisoner of war in Siberia for several years after being captured during the First World War, later tutored him in English and introduced him to set theory and infinite series.

Hungary between the wars was a difficult place for a Jewish student, since the numerus clausus law restricted university admissions. Erdos entered the University of Budapest in 1930 after winning a national examination, an exception granted to competition winners ^[2]. There he fell in with a circle of gifted young mathematicians, including Paul Turan and Tibor Gallai, who met regularly in city parks to trade problems. Several of these friendships turned into lifelong collaborations.

Anyone asking who was Paul Erdos usually hears first about a proof he found as a teenager. In 1932, while still an undergraduate, he published an elegant elementary proof of Chebyshev's theorem, the statement that a prime number always lies between any integer n and its double ^[2]. The result announced a distinctive style: short, clever arguments built from first principles rather than heavy machinery.

He completed his doctorate at the University of Budapest in 1934, at the age of twenty-one, under Leopold Fejer. With antisemitism intensifying in Hungary, he left almost immediately for a fellowship at the University of Manchester in England ^[1]. From Manchester he began the restless pattern that defined the rest of his life, crossing to other universities whenever a problem or a collaborator beckoned.

In 1938 he took a fellowship at the Institute for Advanced Study in Princeton, arriving shortly before the Second World War closed off Europe ^[3]. The war years devastated his family in Hungary; several relatives were murdered in the Holocaust, and his father died of a heart attack in 1942. His mother survived, and their bond remained the closest personal attachment of his life. During the American years he produced fundamental work in number theory and helped found probabilistic methods in combinatorics.

Paul Erdos achievements span an unusual breadth of fields, but number theory anchored his reputation. In 1949 he and Atle Selberg found an elementary proof of the prime number theorem, a result describing how primes thin out among the integers. Mathematicians had long believed such a proof, one avoiding complex analysis, might be impossible, and the achievement drew wide attention along with a lasting dispute over credit between the two men ^[4]. Selberg received the Fields Medal in 1950; Erdos received the Cole Prize of the American Mathematical Society in 1951 for his papers in number theory ^[3].

With Alfred Renyi he built the theory of random graphs in a series of papers beginning in 1959, work that now supports the mathematical study of networks, from the internet to epidemiology ^[5]. He was also a founder of extremal combinatorics and made lasting contributions to Ramsey theory, the study of unavoidable order inside large structures, extending results he had proven with George Szekeres as early as 1935 ^[2].

His output was staggering. Erdos wrote or co-wrote roughly 1,500 mathematical papers, more than any other mathematician of the modern era, with around 500 different co-authors ^[1]. He received the Wolf Prize in 1983, sharing it with Shiing-Shen Chern, and characteristically gave most of the money away ^[3]. He also became famous for attaching small cash prizes, ranging from a few dollars to thousands, to open problems he wanted solved, a practice that still motivates researchers who tackle the surviving Erdos problems.

Erdos owned almost nothing. He never married, never held a permanent job for long, and for decades traveled with one or two suitcases holding all his possessions, staying in the homes of colleagues who fed and housed him while they proved theorems together ^[1]. Friends managed his money, his mail, and often his travel. He gave away most of what he earned, funding prizes, students, and strangers in need.

His speech had its own vocabulary. Children were "epsilons", after the small quantity in mathematical analysis; music was "noise"; to stop doing mathematics was to "die", while actual death was merely "leaving" ^[6]. He called God the "Supreme Fascist", the keeper of a transfinite book containing the most beautiful proof of every theorem, and his highest compliment for an argument was that it came "straight from the Book" ^[6].

He worked nineteen-hour days into old age, sustained by coffee and, from his late fifties onward, by amphetamines prescribed after his mother's death in 1971 left him deeply depressed ^[1]. His mother had traveled with him during her final years, and losing her removed the last fixed point in his personal world. Mathematics filled the space; colleagues recalled him arriving at their doors announcing that his brain was open.

Age barely slowed him. Through his seventies and into his eighties Erdos kept a schedule of conferences and university visits that exhausted much younger companions, often working simultaneously with several collaborators in different rooms ^[1]. His publication rate in his final decades exceeded what most mathematicians manage across a whole career.

Honors accumulated alongside the papers. He was elected to the Hungarian Academy of Sciences in 1956, became an honorary member of the London Mathematical Society in 1973, and was elected a foreign member of the National Academy of Sciences in the United States in 1979 ^[3]. Hungary, which he had left as a young man and visited cautiously during the communist decades, eventually embraced him as a national figure.

He died as he had lived, in motion. On September 20, 1996, while attending a combinatorics conference in Warsaw, Erdos suffered a heart attack and died at the age of eighty-three ^[4]. Only hours earlier he had been working on a problem. He was buried beside his parents in the Kozma Street Jewish cemetery in Budapest.

The most visible trace of his life is the Erdos number, which measures collaborative distance from him: his roughly 500 direct co-authors have Erdos number one, their co-authors have number two, and so on outward through the scientific literature ^[5]. What began as an inside joke among mathematicians became a serious object of study for researchers analyzing collaboration networks, and thousands of scientists across many fields can compute a finite Erdos number.

His problems may matter even more than his theorems. Erdos posed hundreds of precise, stubborn questions across combinatorics, number theory, and geometry, many still carrying his cash bounties, now administered by colleagues and successors ^[2]. Solving an Erdos problem remains a recognized milestone in a mathematician's career, and several of his conjectures have driven whole subfields, including the celebrated 2004 Green-Tao theorem on primes in arithmetic progression, which grew out of a conjecture of Erdos and Turan.

Among the enduring Paul Erdos facts is his redefinition of how mathematics can be done. Before him the subject was largely solitary; his itinerant, radically collaborative style helped normalize joint work, and today most mathematical papers have multiple authors. Paul Hoffman's 1998 biography and a 1993 documentary, both titled with his own phrase about a mathematician being a machine for turning coffee into theorems, carried his story to readers far outside mathematics ^[1]. Any Paul Erdos biography ends the same way: with a man who owned nothing, gave everything away, and left behind one of the largest bodies of work in the history of his science.