HISTORY OF MATHEMATICS – LECTURE 20 – WEDNESDAY 1
ST
DECEMBER
Solved and Unsolved Problems
During this course we have looked at the development of mathematics, and how seemingly intractable
problems were solved or overcome. In particular we have considered how zero was not initially
recognized, that negative solutions to equations were regarded as “absurd”, irrational numbers were
not thought to exist, and complex numbers were not valid.
In the 20th century there were more people employed to do mathematics than in all of history to that
point. This led to many problems being solved, but many others being created, some of which we still
cannot solve. In this lecture we will consider five famous problems that were solved during the second
half of the 20th century, and five others that we still do not know the answer to.
Paul Erd
ő
s
In terms of the number of research papers published, Paul Erd
ő
s (1913
‐
1996) was the most prolific
mathematician of all time, with an output of 1475 papers, written with 511 different collaborators. Born
in Hungary, he lived in home country until the age of 21 before starting a journey that led to him having
no permanent home and no possessions from 1971 onwards, generally just moving to the house of
another mathematician, writing a paper, and then moving on.
While regarded as a world class mathematician, gaining 15 honorary doctorates and membership of the
Royal Society, Erd
ő
s was equally famous for his eccentric behavior, which has inspired three biographies
and a movie. When arriving at the house of another mathematician he would announce “My brain is
open!” and then work on problems, often for up to 20 hours a day, fueled largely by coffee and
amphetamines. Money did not mean a lot to Erd
ő
s, with him once giving away all but $720 of a $50,000
prize that he won to students and those who could solve his challenge problems. However he did accept
a $500 bet that he could not stop taking amphetamines for a month. Erd
ő
s won the bet, but then
complained that mathematical progress had been set back by a month as a result!
Prime Number Theorem (Solved)
In Lecture 16 we looked at how Gauss conjectured the Prime Number Theorem, which we generally
write today as
π
(x)
x/log x.
No proof was found until 1896, when (independently) Jacques Hadamard (1865
‐
1963) and Charles
‐
Jean
de la Vallée
‐
Poussin (1866
‐
1962) came up with long and complicated demonstrations of why the result
holds.
In 1949 Erd
ő
s and Atle Selberg (1917
‐
2007) derived a (more) elementary proof, which was regarded as a
great triumph. There has been controversy ever since as to how much of the work was done by the two
involved, but while Selberg was awarded the Fields Medal in 1950, Erd
ő
s was overlooked, and never
won the award.