## Fulbright.hu

**The Legacy of Paul Erdős**
**Abstract**

*Paul Erdős (Erdős Pál, 1913-1996) was one of the most influential mathematicians of the *
*twentieth century. Erdős’ generation exhibited great talent due in part to the social and *
*cultural changes between 1870 and 1930. In the 1930s, the Jewish Erdős left Hungary due to *
*the worsening political climate, and eventually began a nomadic lifestyle in which he traveled across four continents collaborating with students, teachers, and professors. In popular books, he was described as a wandering ascetic searching only for mathematical truth. Those who knew him best, however, describe him as a compassionate, warm-hearted man. He was addicted to mathematics, but was also aware of current political events and cared deeply for his community of friends. In this paper, I describe the development of mathematical talent in Hungary near the turn of the century, give a biographical sketch of Erdős, and consider his collaborators’ *
*perception of him in light of popular books. Finally, I describe how Hungary continues to nurture mathematical talent, through specialized programs such as the one at Fazekas Gymnasium (Fazekas High School). I also describe the filmmaking projects that I undertook to document these subjects during the past year.*
*Craig Webster: The Legacy of Paul Erdo“s*
**1. Introduction **
**2.1 The Development **
might be necessary to know a bit of a few languages to communicate. Routinely

**of Genius **
In the summer of 2006, I learned about Tibor Frank’s “The Social Construction
During the turn of the twentieth century,
of Genius” to describe factors that may
geographical and political situation has
popular biographies

*My Brain is Open,*
special attention to those factors which
within the culture. Ultimately, this may
by Bruce Schechter, and

*The Man Who *
*Loved Only Numbers,* by Paul Hoffman.
section 2.2, I offer a brief biographical
of Hungary’s most influential scientists,
These books, along with the film

*N is *
*a Number,* were the only sources about
from his colleagues and friends as sources.
Erdős’ life that were readily available.
The books taught me that Erdős was In section 2.3, I consider this portrait in
arguably the most prolific mathematician
in history, but that he was unable to hold
a driver’s license or make a sandwich. He
(see Frank, p. 6 for a detailed list). Since
not been translated into English. I discuss
Toward the end of the nineteenth century,
was a man who devoted his entire life to
influential figures? This phenomenon, (most significantly the Jews) who had
who carried all his belongings in a suit
the best proofs are written in

*the Book*,
which the almighty

*Supreme Fascist* keeps
from humans; who called children

*epsilons*
create social and economic opportunities
and music

*noise*. In these books, Erdős’
stories that he and others tell about his School) attend specialized programs
life. Yet, this year I have spoken to some
of his closest friends and read accounts
by his collaborators, and they emphasize
this past school year, I visited classes and
academic world was less suspicious. Many
warm-hearted nature, social capabilities,
and the Soviets. As a consequence, it has
great mathematicians and scientists from
that period were Jewish, including Johnny
the individual to adapt and assimilate in
Wigner, and Paul Turán (Frank 24-30).

To understand Erdős’ life, it is helpful these programs suggests that Hungary
diversity of ethnic groups, and therefore
has experienced cultural interaction from
cultural framework preceding it. A burst
the Germans, Slavs, Croats, Romas, contributed to the development of young
and many others. This mix has created talent in Budapest. Both innovative
rich lingual environments, in which it and traditional ways of teaching were

*Craig Webster: The Legacy of Paul Erdo“s*
imported through the Gymnasium Szegő, Pólya, Wigner, Teller, Szekeres,
He gained experience and recognition of war in Russia for six years. During that
program. Progressive schools encouraged Lovász, Sós, and Bollabas - to name the
for his part in developing the hydrogen time, Paul’s mother brought him up. She
personal relationships between students most well-known. When asked in 1985
bomb in the Manhattan project, alongside was extraordinarily proud of his abilities
and teachers and experimented with about the “great flowering of Hungarian
von Neumann. He is considered a in mathematics, languages, and history.
new ways of teaching. Prodigies such as mathematics,” Erdős cited Kömal as an
Hungarian-born American, since he lived He grew close to his mother, and they
Johnny von Neumann encountered the interface between experienced professors
more of his life in the U.S. than he did in remained uncommonly close throughout
great math educator László Rátz when he and young students. He said that, “It
Hungary. But both countries take credit his adult life (Sós 205-207, Simonovits
was only ten years old. Immediately, Rátz is where you learned how to solve
for his achievements. Erdős, as always, is and Sós ix-xii).
asked Johnny’s father, Max, if he could problems,” but added that “such things
much more difficult to pin down. He left
tutor Johnny privately. Max agreed, and have more than one reason” (Oláh 3-29).
Hungary when he was 21, but returned In his late teens, Erdős developed his
often to his best friends in Budapest – mathematical talent by solving problems
of charge. By the time Johnny was 17, he The intersection of social and cultural
even during the Cold War. Erdős seemed from Kőmal. Through the journal,
had been at the university for a few years, changes in Budapest helped to create a
to return to Hungary more than any other many of the best mathematicians got to
and many of the professors considered burst of talent in the emerging capital.
place, especially toward the end of his know each other by face and name from
him their peer. The encouragement Hungary’s history of foreign domination
life. Therefore, he would be considered a distance. When these young students
that Rátz gave was paramount to von gradually produced the side effect of
Hungarian. It might be argued that moved to Budapest to study at Pázmány
Neumann’s early success. The informal a creative problem solving ability in
Hungary claims certain famous scientists Péter University (now Eötvös Loránd
relationship between von Neumann and Hungarian culture. Toward the end of the
who lived abroad much of their life. University), they created an informal
Rátz exemplifies the degree to which nineteenth century, Hungary assimilated
Nevertheless, the development of their and wandering group. The group,
accelerated learning can help bright the Jewish community. Jews were drawn
talent is unambiguous: Kömal, the Eötvös consisting of Erdős, Gallai, Grünwald,
students. Rote learning was still common to academic circles, where conversion
competitions, and personal instruction Klein, Szekeres, Turán, Vázsonyi, and
in literature and language, but the to Christianity was less often seen as a
nurtured top students. Nowhere were others, would take excursions through
emergence of industrialization provided social maneuver. New forms of education
these talents developed more acutely the city and trips through the park,
impetus for innovation in the sciences imported from Germany propelled
than in Paul Erdős, one of the greatest discussing mathematics, politics, and
talented students. Kömal and the Eötvös
mathematicians of the twentieth century.
In 1894, the secondary school journal minds, while increased cultural value

**2.2 Biographical Sketch **
Középiskolai Matematikai Lapok (Kömal) was placed on scientific achievement
and the Eötvös math competitions due to industrialization. The result was a

**of Paul Erdős**
began. The journal and the competitions flourishing of mathematical and scientific
intended to stimulate young students talent.
death of his sisters had a great effect on
of the journal contained problems that Yet as Miklós Dezső, the current head of
students were encouraged to solve. They the Rényi Institute points out, it could be
systems. Eventually, they married. Erdős
physics teachers, and they recognized his
would send in their answers, and correct argued that many famous “Hungarian”
talents immediately. Before the age of five,
solutions were published, sometimes scientists owe their achievements as
he could multiply three digit numbers in
with a picture of the solver. Indeed, many much to the United States or England
great Hungarian mathematicians and as to Hungary. For instance, Edward
scientists had their photos printed in Teller left Hungary at the age of 18 and
Paul’s father was captured as a prisoner
Kömal, including Erdős, von Neumann, lived most of his life in the United States.

*Craig Webster: The Legacy of Paul Erdo“s*
of the Hungarian Dénes Kőnig. A few century does one mathematician work
impression that Erdős was some sort of to anything else. He writes, “[Erdős]
years later he gave an elementary proof seriously in so many fields (Halász et.
surreal genius, but in fact he was a solid renounced physical pleasure and material
of Chebeychev’s theorem: that there is al. 55-60). His success was partially due
person, with likes and dislikes, human possessions for an ascetic contemplative
always a prime between

*n *and

*2n *(Aigner to his ability to remain open to others’
frailties, and many lovable qualities.” life, a life devoted to a single narrow
and Ziegner 7-13)

*. *He finished his Ph.D. ideas and problems. He could strike up
Much of the biographical information mission: uncovering mathematical truth”
at 21 and moved to Manchester where mathematical friendships where ever
available to the public is written by (Hoffman 25). In talking with Professor
he studied for four years. He kept in he went with students, teachers, and
outsiders to his life.

*Paul Erdős and his *Miklós Simonovits, a collaborator
good contact with friends and family in professors. Rather than build theorems,

*Mathematics, *however, is written by those of Erdős and graph theorist at the
Budapest, visiting three times a year and he attacked problems, searching for
who knew him best, although the two mathematical research center in Budapest
continuing to collaborate with Turán. their underlying structure. He was
volume set is hard to obtain and is steeped (called the Rényi Institute), I learned that
Their feverish correspondence between renowned for posing problems suited
in mathematical jargon. The first 100 Erdős enjoyed fine dining, and would
1934 and 1940 gives a sense of Erdős’ style to the individual with whom he worked.
pages of volume one fill out the existing go out with his friends when he had
in mathematics and life. His conjectures, In his lectures, he announced general
impression of Erdős by adding the stories the chance. In Simonovits’ estimation,
proofs, and problems are interspersed problems and offered cash rewards for
and opinions of his collaborators. The he was not an ascetic. In fact, he was
with his political and personal concerns solutions. The real reward, however, lay
common theme is that the caring nature well-connected to the world around him.
about the growing anti-Semitic sentiment in discovery of the groundbreaking results
and generous attitude of Erdős stood out Erdős kept up with current political events
that solutions often elicited. During his
and was well-versed in world history. The
Erdős returned to Hungary in 1938. The collaborators and produce nearly 1500
The construction of Erdős’ popular recalls that “[he] had an abiding interest
political climate was so bad that it forced papers.
image has not relied on a combination in political issues in countries all over the
Erdős to leave to the United States. He
of perspectives from people that knew world.” In 1980, while at a conference
him well. The books

*My Brain is Open *in Europe, Erdős stayed up all night to

**2.3 Remembering **
and

*The Man Who Loved Only Numbers, *watch the U.S. Presidential election,
fellowship was not renewed. This is when

**His Legacy**
and the film,

*N is a Number, *were created while Bateman – the American – went to
he began his “nomadic” lifestyle. During
by non-mathematicians outside of his bed. The next morning, Erdős knocked
the next fifty years, he traveled all across Popular accounts of Erdős concentrate
community. These works determine on Bateman’s door and said, “Reagan is
the world, giving lectures, participating on his quirky anecdotes and witty
how the public perceives Erdős. In many your president” (Halász et. al. 9-13).
in conferences and workshops, and remarks. He would recall funny stories
collaborating with other mathematicians in interviews, so much that they became
excite the public about mathematics and Erdős was also devoted to his friends.
canonized. For instance, in almost every
entertain the reader with stories about Both RK Guy and Joel Spencer mention
detailed account of him, he describes the
Erdős’ character. They describe his that he paid attention to the names of
When Erdős was 50 (in 1963), situation of when he was 21 in England
amazing mathematical ability in a way their family members and cared about
celebratory and was faced for the first time to make his
that the layperson can understand. These them on a personal level. During the
paper that surveyed some of Erdős’ own sandwich. This anecdote illustrates
works also recount the events of Erdős’ time of Soviet rule in Hungary, János
accomplishments. A striking fact about his his mother’s early pampering, but perhaps
life, exploring the difficulties that he Surányi recalls how Erdős would bring
work was that he could make significant misleads the public to believe that his
encourntered. However, there are certain back special medications from the West
progress in many areas: number theory, eccentricities were his most exceptional
points that could be represented more to friends and their families behind the
probability theory, graph theory and attribute. There is no doubt that Erdős’
Iron Curtain (Halasz et. al. 21-25, 47-51)

* *
asymptotical combinatorics, constructive life was much different than the norm,
Indeed, Erdős took special care of those
theory of functions, set theory, and but popular accounts tend to exaggerate
In

*The Man Who Loved Only Numbers, *close to him.
set-theoretical topology, theory of series, these details. His collaborator, Paul
and geometry. Rarely in the twentieth Bateman has said, “Some writers give the
his entire life to math and was oblivious

*Craig Webster: The Legacy of Paul Erdo“s*
In

*N is a Number,* Erdős talks about Once I learned about the views of
formal aspect of Hungarian culture. Most from Magyar Television. Ultimately, I
his aversion to sexual stimulation and Erdős’ collaborators

*, *I wanted to create
of the film is a close up of Erdős as he talks would like to make the film available for
explains that he never had a serious a documentary to explain the situation.
and tells stories. It is a sit down discussion, distribution. Since the film is only 50
relationship with a woman. This raises But after speaking with some of his
and there is little attempt to spice it up. minutes, there would need to be more
an ethical question. Is it appropriate to former colleagues at the Rényi Institute, I
Consequently, the film may require some material on the DVD. Kardos created
discuss the sex-life of a mathematician in a experienced resistance. Sós, Erdős’ closest
patience, but in return we see Erdős 70 other films that would be new to the
documentary film? What if he talks about living friend, told me that in her writings
responding to questions spontaneously. English-speaking world. Vera Sós also
it on camera? Naturally, humans want to she had said everything that she wanted
While many accounts of him focus on indicated that there are other Hungarian
know the intimate details of others’ lives. to say about Paul. Twelve years after
what

*happened *in his life, this account also interviews of Erdős yet to be translated.
But it seems that this detail may distract his death, this story did not need to be
focuses on what

*is happening*. It depends Thus, there are many possibilities to fill
revisited by another foreigner who didn’t
less on Erdős retelling stories and more out this DVD, and even create a series
on Erdős expressing his opinion. In one about Hungarian-born mathematicians
During Erdős’ life, he used coffee, I understood the wisdom of this attitude
instance, Erdős discusses the possibilities or based on Kardos’ work.
caffeine pills, and Benzedrine to stay and decided that in fact creating my
awake. In

*The Man Who Loved Only *documentary might not help the situation.
This topic is relevant since we are living The project would require backing

*Numbers, *Hoffman introduces this by While talking to Miklós Simonovits, he
in the future that Erdős talks about. The to distribute. I have written the NSF
writing, “At five foot six, 130 pounds, told me about an interview of Erdős shot
film is also unique because it features with regard to their Informal Science
Erdős had the wizened, cadaverous face of by a Hungarian filmmaker named Kardos
Erdős speaking in his mother tongue. It is Education program. This program
a drug addict, but friends insist that he was István (1942-2006). I had not heard of it.
the only documentation that I am aware in provided partial funding for

*N is a Number *
frail and gaunt before he started taking Did this interview exist in English?
which Erdős speaks Hungarian. Despite and its values seem closely aligned to
amphetamines” (Hoffman 7). Erdős took
its cultural and historical interest, the the benefits of releasing Kardos’ films.
pills on and off throughout his life, and After speaking with the Magyar
film could be criticized in a few ways. For The Templeton Foundation held a
it was not something that he flaunted. Television archive and the National
instance, the questions asked are at times conference in October of 2007 about
Consequently, the discussion may deserve Széchenyi Library, I learned that the
general or speculative. Some may argue Hungarian mathematical talent around
more sensitivity when introduced. When 50 minute piece on Erdős was not in
that it is not “new” enough, since some the turn of the century. I have seen the
Hoffman asked him, Erdős said, “You English. Moreover, Kardos István had
information in the film could be found video documentation of lectures from
shouldn’t have mentioned the stuff about created 70 similar pieces about influential
in existing materials. But the Hungarian the conference via webcast. Perhaps this
Benzedrine. It’s not that you got it wrong. Hungarian mathematicians and scientists,
nature of the production adds a candid footage could be combined to create
It’s just that I don’t want kids who are including 13 Nobel prize winners.
portrait of Uncle Paul that cannot be a documentary about mathematical
thinking about going into mathematics Professor Simonovits gave me a copy of
education, and then released with some of
to think that they have to take drugs to this compilation of interviews of Erdős
During my time in Hungary, I have been Kardos’ films. This work would provide
succeed” (Hoffman 268). In

*N is a Number, *– one shot in 1978, the other in 1988. In
working to make this film available to a Hungarian perspective to the existing
his use of Benzedrine is not included and the film, Kardos and another reporter
English speakers. The most effective material on Erdős, and therefore has
in

*My Brain is Open, *it is mentioned in a ask Erdős questions about his life and
translation, I believe, is subtitles. Subtitles significant cultural value.
more discreet way. The way in which mathematics. He recounts his experience
Erdős’ use of Benzedrine is discussed can as a young man, reflects upon the meaning
would allow the viewer to hear Erdős’
shade the perception of his character. of his work, discusses his philosophy on
There is a fine line between telling the collaborating, and gives his perspective on
complete truth and exploiting the details the future of computers. The austere set,
long pauses, and slow pacing reflect the
help of the Fulbright commission, I am working to gain copyright permission

*Craig Webster: The Legacy of Paul Erdo“s*
Sándor, sat patiently at the front of the
in the U.S. for significant periods of On April 4th, 2008, this year’s competition

**3. Math in the **
time, and he talked about the differences took place, and I was there to film the

**Secondary School **
between Hungary and the U.S. in event. The beginning was relaxed and full
math education. In Hungary, there is a of playful excitement. As time ran out, the
In the 1950s, the International through the window. Twenty minutes
sense of community spirit, and talented pressure and tension increased. Frantic
Mathematics Olympiad (IMO) was later, Sándor strolled the room quietly,
students are encouraged to go deeper participants speed walked to the judges,
introduced, bringing together the top peeking at his students’ work and helping
into mathematics, instead of speed ahead. delivering their solutions. The event
talent from six Soviet countries including when necessary. Sándor’s style is to allow
Whereas in the U.S., the community is brought with it a real sense of enjoyment.
Hungary. Now, more than ninety his students to grapple with the problems,
larger and less closely connected, and the Fazekas brought the NYU computer
countries participate. Students from each in order to encourage independent
math teachers are not trained especially scientist Joel Spencer, as a guest lecturer.
country take two exams. They have four thinking. This practice is designed to
for their job. I hope that the documentary He gave his lecture in Hungarian to a
and a half hours to solve three difficult prepare them for working independently
will bring out these points, so that the crowd of over 90. Most of the audience
problems. Hungary faces giants such at the IMO. They need to develop a
U.S. can learn from Hungary’s example.
as Russia, China, India, and the U.S. self-driven, critical thinking capacity.
attentively as Spencer talked about several
Nevertheless, Hungary repeatedly placed Eventually, Sándor would ask his students
The sense of a small, active community problems involving asymptotes and graph
within the top ranks of the competition to share their solutions on the chalkboard.
was evident in the “Fazekas Napok” theory. It seemed that everyone in the
and many Hungarian students walk away The students explained their discoveries,
(Fazekas Days), an informal math audience wanted to attend the two hour
competition involving over 100 students. session, as classes had been cancelled for Teams of seven are given 21 questions, the occasion.

Fazekas Gymnasium, located in the Hungarian mathematics education is a
and they have 90 minutes to solve as many
8th District in Budapest, is one of the century old tradition, which continues
problems as they can. A correct answer I may have stumbled upon yet another
secondary school which has specialized today. More than 15,000 students each
wins the team between 20 to 40 points. reason that high level mathematics is
math programs for talented students. year participate in the competitions
The team loses 10 points for an incorrect so good in Hungary. Not only is the
Toward the end of February, I began hosted by the Bolyai Society (including
answer. Pastries and soft drinks are given community small, but its members
to attend the informal preparations the former Eötvös competition). Many
to the teams, who sit at circular tables and are friendly and helpful. Professor
for the IMO. I could understand only students still greet the new publication of
work intensely on the problems. Once a Simonovits introduced me to the head
bits of Hungarian, but could see how Kőmal with excitement and send answers
team has a solution, the team’s runner coach of Hungary’s IMO team – József
comfortable these 17 and 18 year-olds each month. The Fazekas program is
delivers it to a cadre of judges, mostly Pelikán – who then introduced me to
were when they solved problems at especially active, and similar programs
first or second year students from the Sándor Dobos. Teachers such as Sándor
the chalkboard. I had rarely seen such exist throughout Hungary. I am currently
university (many of whom went to Fazekas are relentless in their generosity. He has
confidence at my high school in the creating a documentary film about
the year before). Each answer is a single introduced me to other teachers and
U.S., even when we solved much easier secondary school math education, paying
integer between 0 and 10,000 written on a educators, who in turn have invited me for
problems. Of course, the Hungarian math particular attention to the development
small slip of paper. The judges grade them an interview or into their classrooms to
students who participated in the voluntary of bright students. I interviewed Dobos
on the spot. The scoreboard changes in observe and film. Each person that I talk to
Friday afternoon preparation were no Sándor and some of his students, and
real-time, as teams turn in answers, and speaks highly of another, and this friendly
ordinary students. They were some of the filmed both advanced and regular-track
the teams battle each other until the very spirit is conducive to collaboration. As
last minute. The competition encourages the Fazekas teacher Pátaki János told
collaboration and cross-checking between me, the United States may be good at
After giving a few problems to the researcher in combinatorics and a
students. Regardless of whether they win, creating strong mathematical institutions
students, the teacher (and deputy coach member of the first ever specialized math
of the Hungarian IMO team), Dobos program at Fazekas in 1962. He has lived

*Craig Webster: The Legacy of Paul Erdo“s*
how to create a community spirit, where Paul Erdős was one of the greatest
math students. The tight-knit math Bibliography
the best students are exposed to their mathematicians of the twentieth century.
community and tradition of Kőmal and I have used quotes and ideas from interviews with Laszlo Lozász,
peers so that they can learn and develop His character has been recorded in
other competitions bring bright students Miklos Simonovits, and Joel Spencer, which were conducted
together. Given this trend, the legacy of popular biographies by American authors
together throughout their development. in 2008 at Eötvös Loránd University, the Rényi Insitute and
developing talented youth in mathematics who accentuate funny stories about him
Perhaps more attention should be paid Fazekas Gymnasium.
will continue, independent of political more than his genuine kindness. While he
problems or lack of funding. The was addicted to math, he also cared about
in mathematics. Teachers such as Sándor Albers, DJ and Alexanderson, GJ. “Interview with P. Erdős.”
Hungarian math community remains the people around him and paid attention
Dobos encourage women to participate,

*Mathematical People*, Birkhauser, Boston: 1985.
strong and healthy despite the radical to history and politics. When I spoke with
changes of the past century, and appears his former collaborators, many approved
mathematics as a very “womanly” subject. Aigner, Martin and Ziegler, Günter.

*Proofs from the Book. *
These social morays may change, but the Springer, New York: 2004.
legacy of outstanding mathematical talent
of Erdős and his contemporaries will Csicery, George.

*N is a Number. *Zala Films, 1993.

**4. Conclusion**
István shows Erdős talking in Hungarian
about his life. I hope that a translation
Frank, Tibor. “The Social Construction of Genius,” from the
During this past year I learned about of the film would provide a fresh, new
Conference “Budapest: the Golden Years,” Princeton, 2007.

the golden age of Hungary. The sharp increase of talent in math and science perspective that is less adulterated than
Halász, G., Lovász, L., Simonovits, M. and Sós, V.T. (Eds.)

*Paul *
was in large part due to a confluence of only documentation (that I am aware

*Erdős and his Mathematics, *Springer: Budapest, 2002.

cultural and social developments linked to the newly emerging capital Budapest, of) in which Erdős speaks in his mother
Hoffman, Paul.

*The Man Who Loved Only Numbers. *New York,

* *
innovative educational practices imported the calm and formal nature of Hungarian
from Germany, and the assimilation of the Jewish community. The Kömal journal culture. Moreover, the Kardos interview
Oláh, V. “Centennial Issue of Kömal,” János Bolyai
and the Eötvös competitions encouraged giving a chance for Erdős to express his
students to solve challenging questions and rewarded young talent. Due to the opinion about modern issues such as how
Schechter, Bruce.

*My Brain is Open*. Touchstone, New York:

* *
political climate in Hungary around I am working on gaining copyright
World War II, many of these scientists were forced to leave for the United States. permission to add English subtitles, while
I create a subtitle track. I am also finding a
Simonovits, M. and Sós, V. T.

*Recent Trends in Combinatorics. *
It could be said that some of the great way for the film to be distributed.
Forward. “Paul Erdős: The Man and the Mathematician,” CUP,
scientists are as much American as they are
Hungarian, since their most celebrated work occurred in the U.S. But the debt Simultaneously, I am creating a
Sós, V. T. “Paul Erdős”,

*Aequationes Math. ***54 **(1997) 205 – 220.

they owe to Hungary is undeniable. It is clear that the innovative education and secondary school education in Hungary exciting competitions in Hungary helped that should be finished in May. I have to develop the talent of many young observed classes, interviewed students, students, some of which went on to do teachers, and professors, and filmed great works. This legacy continues today competitions. The end product will through specialized secondary school show some of the keys to Hungary’s programs such as Fazekas.

Source: http://www.fulbright.hu/book5/craigwebster.pdf

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