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HISTORY OF MATHEMATICS – LECTURE 16 – WEDNESDAY 3
RD
NOVEMBER
Carl Friedrich Gauss
Known as the “Prince of Mathematics“, Carl Friedrich Gauss (1777
‐
1855) is universally regarded as one
of the greatest mathematicians of all time. Although hailing from a poor family in Brunswick, Germany,
the talent of Gauss was obvious from a young age, and he regularly astonished his school masters with
his knowledge.
His extraordinary ability as a child caused the Duke of Brunswick to finance Gauss through preparatory
school, and then the University of Göttingen from 1795
‐
1798. Whilst there Gauss made a series of
interesting discoveries, generalizing the polygons which can be constructed using straight edge and
compass (an ancient problem that was thought to have been solved), and proving the quadratic
reciprocity law while still only 18. Later that year he proved that every positive integer can be written as
the sum of three triangular numbers, and gave a version of the prime number theorem, which we will
consider below. He also invented modular arithmetic, which led to the publication in 1801 of his most
famous work,
Disquisitiones Arithmetica
(Arithmetic Investigations).
Prime Number Theorem
We proved in Lecture 5 that there are infinitely many prime numbers. So since we cannot determine all
of them, mathematicians have long been interested in how they are distributed amongst the natural
numbers. No exact answer has ever been found, and so instead we try to approximate how many primes
exist between 1 and any given number. We (confusingly) use
π
to denote this, and so
π
(10) = 4, since
there are 4 prime numbers less than 10. Given that there are infinitely many prime numbers,
π
(x) must
tend to infinity as x does, but the number of primes does tend to decrease as x gets larger (see below).
x
π
(x)
Number of new primes
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Gauss formed this table when was 15, and conjectured that the number of primes less than or equal to
the natural number x is given approximately by
ଵ
୪୭
௫
ଶ
݀݊
.
While not a perfect approximation, it is very accurate (for example when x = 3,000,000,
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 Spring '08
 EVINSON
 Math, Calculus

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