Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
s of the 20th century. pursued and proposed problems in
discrete mathematics
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...
,
graph theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
,
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
,
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
,
approximation theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
,
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, and
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
.
Much of his work centered on discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in R ...
, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously
open problem
In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is kno ...
s, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed.
He was known both for his social practice of mathematics, working with more than 500 collaborators, and for his
eccentric lifestyle; ''
Time
Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
'' magazine called him "The Oddball's Oddball". He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathematicians. He devoted his waking hours to mathematics, even into his later years; he died at a mathematics conference in
Warsaw
Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at ...
in 1996.
Erdős's prolific output with co-authors prompted the creation of the
Erdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships.
Life
Paul Erdős was born on 26 March 1913, in
Budapest
Budapest is the Capital city, capital and List of cities and towns of Hungary, most populous city of Hungary. It is the List of cities in the European Union by population within city limits, tenth-largest city in the European Union by popul ...
,
Austria-Hungary
Austria-Hungary, also referred to as the Austro-Hungarian Empire, the Dual Monarchy or the Habsburg Monarchy, was a multi-national constitutional monarchy in Central Europe#Before World War I, Central Europe between 1867 and 1918. A military ...
, the only surviving child of Anna (
née
The birth name is the name of the person given upon their birth. The term may be applied to the surname, the given name or to the entire name. Where births are required to be officially registered, the entire name entered onto a births registe ...
Wilhelm) and Lajos Erdős (né Engländer).
His two sisters, aged three and five, both died of
scarlet fever a few days before he was born. His parents, both
Jewish
Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...
, were high school mathematics teachers. His fascination with mathematics developed early. He was raised partly by a German governess
[ because his father was held captive in ]Siberia
Siberia ( ; , ) is an extensive geographical region comprising all of North Asia, from the Ural Mountains in the west to the Pacific Ocean in the east. It has formed a part of the sovereign territory of Russia and its predecessor states ...
as an Austro-Hungarian prisoner of war
A prisoner of war (POW) is a person held captive by a belligerent power during or immediately after an armed conflict. The earliest recorded usage of the phrase "prisoner of war" dates back to 1610.
Belligerents hold prisoners of war for a ...
during 1914–1920, causing his mother to have to work long hours to support their household. His father had taught himself English while in captivity but mispronounced many words. When Lajos later taught his son to speak English, Paul learned his father's pronunciation, which he continued to use for the rest of his life.
He taught himself to read through mathematics texts that his parents left around in their home. By the age of five, given a person's age, he could calculate in his head how many seconds they had lived. Due to his sisters' deaths, he had a close relationship with his mother, with the two of them reportedly sharing the same bed until he left for college.
When he was 16, his father introduced him to two subjects that would become lifetime favourites—infinite series
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
and set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
. In high school, Erdős became an ardent solver of the problems that appeared each month in '' KöMaL'', the "Mathematical and Physical Journal for Secondary Schools".
Erdős began studying at the University of Budapest when he was 17 after winning a national examination. At the time, admission of Jews to Hungarian universities was severely restricted under the '' numerus clausus''. By the time he was 20, he had found a proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a co ...
for Bertrand's postulate
In number theory, Bertrand's postulate is the theorem that for any integer n > 3, there exists at least one prime number p with
:n < p < 2n - 2.
A less restrictive formulation is: for every , there is always at least one ...
. In 1934, at the age of 21, he was awarded a doctorate in mathematics. Erdős's thesis advisor was Lipót Fejér, who was also the thesis advisor for John von Neumann
John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
, George Pólya
George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributi ...
, and Pál Turán. He took up a post-doctoral fellowship at Manchester
Manchester () is a city and the metropolitan borough of Greater Manchester, England. It had an estimated population of in . Greater Manchester is the third-most populous metropolitan area in the United Kingdom, with a population of 2.92&nbs ...
, as Jews in Hungary were suffering oppression under the authoritarian regime
Authoritarianism is a political system characterized by the rejection of political plurality, the use of strong central power to preserve the political ''status quo'', and reductions in democracy, separation of powers, civil liberties, and ...
. While there, he met G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and Stanisław Ulam
Stanisław Marcin Ulam ( ; 13 April 1909 – 13 May 1984) was a Polish and American mathematician, nuclear physicist and computer scientist. He participated in the Manhattan Project, originated the History of the Teller–Ulam design, Telle ...
.
Because he was Jewish, Erdős decided Hungary was dangerous and left the country, relocating to the United States in 1938. Many members of Erdős's family, including two of his aunts, two of his uncles, and his father, died in Budapest during World War II. His mother was the only one that survived. He was living in America and working at the Institute for Advanced Study
The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
in Princeton at the time. However, his fellowship at Princeton only got extended by 6 months rather than the expected year due to Erdős not conforming to the standards of the place; they found him "uncouth and unconventional".
Described by his biographer, Paul Hoffman, as "probably the most eccentric mathematician in the world," Erdős spent most of his adult life living out of a suitcase. Except for some years in the 1950s, when he was not allowed to enter the United States based on the accusation that he was a Communist sympathizer, his life was a continuous series of going from one meeting or seminar to another. During his visits, Erdős expected his hosts to lodge him, feed him, and do his laundry, along with anything else he needed, as well as arrange for him to get to his next destination.
Ulam left his post at the University of Wisconsin–Madison
The University of Wisconsin–Madison (University of Wisconsin, Wisconsin, UW, UW–Madison, or simply Madison) is a public land-grant research university in Madison, Wisconsin, United States. It was founded in 1848 when Wisconsin achieved st ...
in 1943 to work on the Manhattan Project
The Manhattan Project was a research and development program undertaken during World War II to produce the first nuclear weapons. It was led by the United States in collaboration with the United Kingdom and Canada.
From 1942 to 1946, the ...
in Los Alamos, New Mexico
Los Alamos (, meaning ''The Poplars'') is a census-designated place in Los Alamos County, New Mexico, United States, that is recognized as one of the development and creation places of the Nuclear weapon, atomic bomb—the primary objective of ...
with other mathematicians and physicists. He invited Erdős to join the project, but the invitation was withdrawn when Erdős expressed a desire to return to Hungary after the war.
On 20 September 1996, at the age of 83, he had a heart attack
A myocardial infarction (MI), commonly known as a heart attack, occurs when Ischemia, blood flow decreases or stops in one of the coronary arteries of the heart, causing infarction (tissue death) to the heart muscle. The most common symptom ...
and died while attending a conference in Warsaw
Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at ...
. These circumstances were close to the way he wanted to die. He once said,
Erdős never married and had no children. He is buried next to his mother and father in the Jewish Kozma Street Cemetery in Budapest. For his epitaph
An epitaph (; ) is a short text honoring a deceased person. Strictly speaking, it refers to text that is inscribed on a tombstone or plaque, but it may also be used in a figurative sense. Some epitaphs are specified by the person themselves be ...
, he suggested "I've finally stopped getting dumber." (Hungarian: ''"Végre nem butulok tovább"'').
Erdős's name contains the Hungarian letter "ő" ("o" with double acute accent
The double acute accent () is a diacritic mark of the Latin and Cyrillic scripts. It is used primarily in Hungarian or Chuvash, and consequently it is sometimes referred to by typographers as hungarumlaut. The signs formed with a regular umlau ...
), but is often incorrectly written as ''Erdos'' or ''Erdös'' either "by mistake or out of typographical necessity".
Career
In 1934, Erdős moved to Manchester
Manchester () is a city and the metropolitan borough of Greater Manchester, England. It had an estimated population of in . Greater Manchester is the third-most populous metropolitan area in the United Kingdom, with a population of 2.92&nbs ...
, England, to be a guest lecturer. In 1938, he accepted his first American position as a scholarship holder at the Institute for Advanced Study
The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
, Princeton, New Jersey, for the next ten years. Despite outstanding papers with Mark Kac and Aurel Wintner on probabilistic number theory, Pál Turán in approximation theory, and Witold Hurewicz on dimension theory, his fellowship was not continued, and
Erdős was forced to take positions as a wandering scholar at UPenn, Notre Dame, Purdue, Stanford, and Syracuse. He would not stay long in one place, instead traveling among mathematical institutions until his death.
As a result of the Red Scare and McCarthyism
McCarthyism is a political practice defined by the political repression and persecution of left-wing individuals and a Fear mongering, campaign spreading fear of communist and Soviet influence on American institutions and of Soviet espionage i ...
, in 1954, the United States Immigration and Naturalization Service denied Erdős, a Hungarian citizen, a re-entry visa into the United States. Teaching at the University of Notre Dame
The University of Notre Dame du Lac (known simply as Notre Dame; ; ND) is a Private university, private Catholic research university in Notre Dame, Indiana, United States. Founded in 1842 by members of the Congregation of Holy Cross, a Cathol ...
at the time, Erdős could have chosen to remain in the country. Instead, he packed up and left, albeit requesting reconsideration from the U.S. Immigration Services at periodic intervals. At some point he moved to live in Israel. He was given a position for three months at the Hebrew University in Jerusalem, and then a "permanent visiting professor" position at the Technion.
Hungary
Hungary is a landlocked country in Central Europe. Spanning much of the Pannonian Basin, Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to the northeast, Romania to the east and southeast, Serbia to the south, Croatia and ...
at the time was under the Warsaw Pact
The Warsaw Pact (WP), formally the Treaty of Friendship, Co-operation and Mutual Assistance (TFCMA), was a Collective security#Collective defense, collective defense treaty signed in Warsaw, Polish People's Republic, Poland, between the Sovi ...
with the Soviet Union
The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...
. Although Hungary limited the freedom of its own citizens to enter and exit the country, in 1956 it gave Erdős the exclusive privilege of being allowed to enter and exit the country as he pleased.
In 1963, the United States Immigration and Naturalization Service granted Erdős a visa, and he resumed teaching at and traveling to American institutions. Ten years later, in 1973, the 60-year-old Erdős voluntarily left Hungary.
During the last decades of his life, Erdős received at least fifteen honorary doctorates. He became a member of the scientific academies of eight countries, including the U.S. National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
and the UK Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
. He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1977. Shortly before his death, he renounced his honorary degree from the University of Waterloo
The University of Waterloo (UWaterloo, UW, or Waterloo) is a Public university, public research university located in Waterloo, Ontario, Canada. The main campus is on of land adjacent to uptown Waterloo and Waterloo Park. The university also op ...
over what he considered to be unfair treatment of colleague Adrian Bondy.
Mathematical work
Erdős was one of the most prolific publishers of papers in mathematical history, comparable only with Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
; Erdős published more papers, mostly in collaboration with other mathematicians, while Euler published more pages, mostly by himself. Erdős wrote around 1,525 mathematical articles in his lifetime, mostly with co-authors. He strongly believed in and practiced mathematics as a social activity, having 511 different collaborators in his lifetime.
In his mathematical style, Erdős was much more of a "problem solver" than a "theory developer" (see "The Two Cultures of Mathematics" by Timothy Gowers
Sir William Timothy Gowers, (; born 20 November 1963) is a British mathematician. He is the holder of the Combinatorics chair at the Collège de France, a director of research at the University of Cambridge and a Fellow of Trinity College, Camb ...
for an in-depth discussion of the two styles, and why problem solvers are perhaps less appreciated). Joel Spencer states that "his place in the 20th-century mathematical pantheon is a matter of some controversy because he resolutely concentrated on particular theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s and conjectures throughout his illustrious career." Erdős never won the Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
(the highest mathematical prize available during his lifetime), nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes. He did win the 1983/84 Wolf Prize
The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for "achievements in the interest of mankind and friendly relations among people ... irrespective of natio ...
, "for his numerous contributions to number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
, probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
, set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
and mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, and for personally stimulating mathematicians the world over". In contrast, the works of the three winners after were recognized as "outstanding", "classic", and "profound", and the three before as "fundamental" or "seminal".
Of his contributions, the development of Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in R ...
and the application of the probabilistic method especially stand out. Extremal combinatorics owes to him a whole approach, derived in part from the tradition of analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
. Erdős found a proof for Bertrand's postulate
In number theory, Bertrand's postulate is the theorem that for any integer n > 3, there exists at least one prime number p with
:n < p < 2n - 2.
A less restrictive formulation is: for every , there is always at least one ...
which proved to be far neater than Chebyshev's original one. He also discovered the first elementary proof for the prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by p ...
, along with Atle Selberg
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded ...
. However, the circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between Erdős and Selberg. Erdős also contributed to fields in which he had little real interest, such as topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, where he is credited as the first person to give an example of a totally disconnected topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
that is not zero-dimensional, the Erdős space.
Erdős's problems
Erdős had a reputation for posing new problems as well as solving existing ones: Ernst Strauss called him "the absolute monarch of problem posers". Throughout his career, Erdős would offer payments for solutions to unresolved problems. These ranged from $25 for problems that he felt were just out of the reach of the current mathematical thinking (both his and others) up to $10,000 for problems that were both difficult to attack and mathematically significant. Some of these problems have since been solved, including the most lucrative – Erdős's conjecture on prime gap
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-st and the ''n''-th prime numbers, i.e.,
:g_n = p_ - p_n. ...
s was solved in 2014, and the $10,000 paid.
There are thought to be at least a thousand remaining unsolved problems, though there is no official or comprehensive list. The offers remained active despite Erdős's death; Ronald Graham was the (informal) administrator of solutions, and a solver could receive either an original check signed by Erdős before his death (for memento only, which cannot be cashed) or a cashable check from Graham. British mathematician Thomas Bloom started a website dedicated to Erdős's problems in 2024.
Perhaps the most mathematically notable of these problems is the Erdős conjecture on arithmetic progressions:
If true, it would solve several other open problems in number theory, although one main implication of the conjecture, that the prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s contain arbitrarily long arithmetic progressions, has since been proved independently as the Green–Tao theorem. The payment for the solution of the problem is currently worth US$5,000.
The most familiar problem with an Erdős prize is likely the Collatz conjecture
The Collatz conjecture is one of the most famous List of unsolved problems in mathematics, unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer ...
, also called the 3''N'' + 1 problem. Erdős offered $500 for a solution.
Collaborators
Erdős's most frequent collaborators include Hungarian mathematicians András Sárközy (62 papers) and András Hajnal (56 papers), and American mathematician Ralph Faudree (50 papers). Other frequent collaborators were the following:
* Richard Schelp (42 papers)
* Cecil C. Rousseau (35 papers)
* Vera T. Sós (35 papers)
* Alfréd Rényi (32 papers)
* Pál Turán (30 papers)
* Endre Szemerédi (29 papers)
* Ron Graham (28 papers)
* Stefan Burr (27 papers)
* Carl Pomerance (23 papers)
* Joel Spencer (23 papers)
* János Pach (21 papers)
* Miklós Simonovits (21 papers)
* Ernst G. Straus (20 papers)
* Melvyn B. Nathanson (19 papers)
* Jean-Louis Nicolas (19 papers)
* Richard Rado (18 papers)
* Béla Bollobás (18 papers)
* Eric Charles Milner (15 papers)
* András Gyárfás (15 papers)
* John Selfridge (14 papers)
* Fan Chung (14 papers)
* Richard R. Hall (14 papers)
* George Piranian (14 papers)
* István Joó (12 papers)
* Zsolt Tuza (12 papers)
* A. R. Reddy (11 papers)
* Vojtěch Rödl (11 papers)
* Pál Révész (10 papers)
* Zoltán Füredi (10 papers)
For other co-authors of Erdős, see the list of people with Erdős number 1 in List of people by Erdős number.
Erdős number
Because of his prolific output, friends created the Erdős number as a tribute. An Erdős number describes a person's degree of separation from Erdős himself, based on their collaboration with him, or with another who has their own Erdős number. Erdős alone was assigned the Erdős number of 0 (for being himself), while his immediate collaborators could claim an Erdős number of 1, their collaborators have Erdős number at most 2, and so on. Approximately 200,000 mathematicians have an assigned Erdős number, and some have estimated that 90 percent of the world's active mathematicians have an Erdős number smaller than 8 (not surprising in light of the small-world phenomenon). Due to collaborations with mathematicians, many scientists in fields such as physics, engineering, biology, and economics also have Erdős numbers.
Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers.[ Original Spanish version in ''Rev. Acad. Colombiana Cienc. Exact. Fís. Natur.'' 23 (89) 563–582, 1999, .] For example, the roughly 268,000 mathematicians with a known Erdős number have a median value of 5. In contrast, the median Erdős number of Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
ists is 3. As of 2015, approximately 11,000 mathematicians have an Erdős number of 2 or lower. Collaboration distances will necessarily increase over long time scales, as mathematicians with low Erdős numbers die and become unavailable for collaboration. The American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
provides a free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews
''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science.
The AMS also pu ...
catalogue.
The Erdős number was most likely first defined by Casper Goffman, an analyst whose own Erdős number is 2; Goffman co-authored with mathematician Richard B. Darst, who co-authored with Erdős. Goffman published his observations about Erdős's prolific collaboration in a 1969 article titled "And what is your Erdős number?"
Jerry Grossman has written that it could be argued that Baseball Hall of Famer Hank Aaron can be considered to have an Erdős number of 1, because they both autographed the same baseball for Carl Pomerance when Emory University
Emory University is a private university, private research university in Atlanta, Georgia, United States. It was founded in 1836 as Emory College by the Methodist Episcopal Church and named in honor of Methodist bishop John Emory. Its main campu ...
awarded them honorary degrees on the same day. Erdős numbers have also been proposed for an infant, a horse, and several actors.
Personality
Possessions meant little to Erdős; most of his belongings would fit in a suitcase, as dictated by his itinerant lifestyle. Awards and other earnings were generally donated to people in need and various worthy causes. He spent most of his life traveling between scientific conferences, universities, and the homes of colleagues all over the world. He earned enough in stipends from universities as a guest lecturer, and from various mathematical awards, to fund his travels and basic needs; money left over he used to fund cash prizes for proofs of "Erdős's problems" (see above). He would typically show up at a colleague's doorstep and announce "my brain is open", staying long enough to collaborate on a few papers before moving on a few days later. In many cases, he would ask the current collaborator about whom to visit next.
His colleague Alfréd Rényi said, "A mathematician is a machine for turning coffee
Coffee is a beverage brewed from roasted, ground coffee beans. Darkly colored, bitter, and slightly acidic, coffee has a stimulating effect on humans, primarily due to its caffeine content, but decaffeinated coffee is also commercially a ...
into theorems", and Erdős drank copious quantities; this quotation is often attributed incorrectly to Erdős, but Erdős ascribed it to Rényi. After his mother's death in 1971 he started taking antidepressants and amphetamines, despite the concern of his friends, one of whom ( Ron Graham) bet him $500 that he could not stop taking them for a month. Erdős won the bet but complained that it impacted his performance: "You've showed me I'm not an addict. But I didn't get any work done. I'd get up in the morning and stare at a blank piece of paper. I'd have no ideas, just like an ordinary person. You've set mathematics back a month." After he won the bet, he promptly resumed his use of Ritalin and Benzedrine.
He had his own idiosyncratic vocabulary; although an agnostic atheist, he spoke of "The Book", a visualization of a book in which God
In monotheistic belief systems, God is usually viewed as the supreme being, creator, and principal object of faith. In polytheistic belief systems, a god is "a spirit or being believed to have created, or for controlling some part of the un ...
had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, "You don't have to believe in God, but you should believe in ''The Book''." He doubted the existence of God. He playfully nicknamed him the SF (for "Supreme Fascist"), accusing him of hiding his socks and Hungarian passport
A passport is an official travel document issued by a government that certifies a person's identity and nationality for international travel. A passport allows its bearer to enter and temporarily reside in a foreign country, access local aid ...
s, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof he would exclaim, "This one's from ''The Book''!" This later inspired a book titled ''Proofs from the Book
''Proofs from THE BOOK'' is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathemat ...
''.
Other idiosyncratic elements of Erdős's vocabulary include:
* Children were referred to as "epsilon
Epsilon (, ; uppercase , lowercase or ; ) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel or . In the system of Greek numerals it also has the value five. It was derived from the Phoenic ...
s", because in mathematics, particularly calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, an arbitrarily small positive quantity is commonly denoted by the Greek letter (ε).
* Women were "bosses" who "captured" men as "slaves" by marrying them. Divorced men were "liberated".
* People who stopped doing mathematics had "died", while people who died had "left".
* Alcoholic drinks were "poison".
* Music, except classical music, was "noise".
* To be considered a hack was to be a "Newton".
* To give a mathematical lecture was "to preach".
* Mathematical lectures were "sermons".
* To give an oral exam to students was "to torture" them.
He gave nicknames to many countries, examples being: the U.S. was "samland" (after Uncle Sam) and the Soviet Union was "joedom" (after Joseph Stalin
Joseph Vissarionovich Stalin (born Dzhugashvili; 5 March 1953) was a Soviet politician and revolutionary who led the Soviet Union from 1924 until Death and state funeral of Joseph Stalin, his death in 1953. He held power as General Secret ...
). He claimed that Hindi
Modern Standard Hindi (, ), commonly referred to as Hindi, is the Standard language, standardised variety of the Hindustani language written in the Devanagari script. It is an official language of India, official language of the Government ...
was the best language, because words for old age ('' bud̩d̩hā'') and stupidity ('' buddhū'') sounded almost the same.
Signature
Erdős signed his name "Paul Erdos P.G.O.M." When he became 60, he added "L.D.", at 65 "A.D.", at 70 "L.D." (again), and at 75 "C.D."
* P.G.O.M. represented "Poor Great Old Man"
* The first L.D. represented "Living Dead"
* A.D. represented "Archaeological Discovery"
* The second L.D. represented "Legally Dead"
* C.D. represented "Counts Dead"
Legacy
Books and films
Erdős is the subject of at least three books: two biographies ( Hoffman's '' The Man Who Loved Only Numbers'' and Schechter's ''My Brain is Open'', both published in 1998) and a 2013 children's picture book by Deborah Heiligman (''The Boy Who Loved Math: The Improbable Life of Paul Erdős'').
He is also the subject of George Csicsery's biographical documentary film '' N is a Number: A Portrait of Paul Erdős,'' made while he was still alive.
Astronomy
In 2021 the minor planet
According to the International Astronomical Union (IAU), a minor planet is an astronomical object in direct orbit around the Sun that is exclusively classified as neither a planet nor a comet. Before 2006, the IAU officially used the term ''minor ...
(asteroid
An asteroid is a minor planet—an object larger than a meteoroid that is neither a planet nor an identified comet—that orbits within the Solar System#Inner Solar System, inner Solar System or is co-orbital with Jupiter (Trojan asteroids). As ...
) 405571 (temporarily designated 2005 QE87) was formally named "Erdőspál" to commemorate Erdős, with the citation describing him as "a Hungarian mathematician, much of whose work centered around discrete mathematics. His work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics." The naming was proposed by "K. Sárneczky, Z. Kuli" (Kuli being the asteroid's discoverer).
Arachnology
In 2025 British arachnologists Danniella Sherwood and R. C. Gallon described ''Heterothele erdosi'' Sherwood & Gallon, 2025, a new species of tarantula
Tarantulas comprise a group of large and often hairy spiders of the family Theraphosidae. , 1,100 species have been identified, with 166 genera. The term "tarantula" is usually used to describe members of the family Theraphosidae, although ...
from Nigeria
Nigeria, officially the Federal Republic of Nigeria, is a country in West Africa. It is situated between the Sahel to the north and the Gulf of Guinea in the Atlantic Ocean to the south. It covers an area of . With Demographics of Nigeria, ...
, named in honour of Erdős. [Sherwood, D., Gallon, R. C. (2025). "''Heterothele'' ''erdosi'', a new species of ''Heterothele'' Karsch, 1879 from Nigeria (Araneae: Theraphosidae)". ''Natural Somogyiensis'' 45: 51–56]
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See also
* – including conjectures, numbers, prizes, and theorems
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References
Sources
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Further reading
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External links
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Searchable collection of (almost) all papers of Erdős
Database of problems proposed by Erdős
''The Erdös Number Project''
at Oakland University
Oakland University (OU or Oakland) is a public university, public research university in Auburn Hills, Michigan, Auburn Hills and Rochester Hills, Michigan, United States. Founded in 1957 through a donation of Matilda Dodge Wilson and husband ...
The Man Who Loved Only Numbers
public lecture by Paul Hoffman
"Open problems of Paul Erdős in graph theory"
by Fan Chung
{{DEFAULTSORT:Erdos, Paul
1913 births
1996 deaths
20th-century Hungarian Jews
20th-century Hungarian mathematicians
Academics of the Victoria University of Manchester
Burials at Kozma Street Cemetery
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Graph theorists
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