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Euclid of Alexandria, II: Number Theory
Euclid of Alexandria, II:
Number Theory
Waseda University, SILS,
History of Mathematics
Euclid of Alexandria, II: Number Theory
Outline
Introduction
Euclid’s number theory
The overall structure
Definitions for number theory
Theory of prime numbers
Properties of primes
Infinitude of primes
Euclid of Alexandria, II: Number Theory
Introduction
Concepts of number
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The natural numbers is the set N “ t1, 2, 3, . . . u.
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The whole numbers is the set W “ t0, 1, 2, 3, . . . u.
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The integers is the set of positive and negative whole
numbers Z “ t0, 1, ´1, 2, ´2, . . . u.
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The rational numbers is the set Q, of numbers of the form
where p, q P Z,1 but q ‰ 0.
p{q,
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The real numbers, R, is the set of all the values mapped to
the points of the number line. (The definition is tricky.)
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An irrational number is a number that belongs to the reals,
but is not rational.
1
The symbol P means “in the set of,” or “is an element of.”
Euclid of Alexandria, II: Number Theory
Introduction
The Greek concept of number
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Greek number theory was exclusively interested in natural
numbers.
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In fact, the Greek also did not regard “1” as a number, but
rather considered it the unit by which other numbers are
numbered (or measured).
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We can define Greek natural numbers as G “ t2, 3, 4, . . . u.
(But we can do most Greek number theory with N, so we
will generally use this set, for simplicity.)
Euclid of Alexandria, II: Number Theory
Introduction
Number theory before Euclid
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The semi-legendary Pythagorus himself and other
Pythagoreans are attributed with a fascination with
numbers and with the development of a certain “pebble
arithmetic” which studied the mathematical properties of
numbers that correspond to certain geometry shapes
(figurate numbers).
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Philolaus of Croton (late 5th, earth 4th BCE) is attributed
with some numerological speculations related to music
theory and cosmology.
§
Archytus of Tarentum (4th BCE) is attributed with some
theorems of number theory, most of which are directly
applicable to ancient music theory.
Euclid of Alexandria, II: Number Theory
Euclid’s number theory
The overall structure
Elements VII–IX
As in earlier books, Euclid probably based much of his work on
the discoveries of others, but the organization and presentation
was his own.
§
Book VII: numbers and proportions, theory of divisors,
theory of least common multiples
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Book VIII: theory of figurate numbers, mean proportionals
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Book IX: numbers and proportions, theory of prime
numbers, theory of even and odd, perfect numbers2
2
A perfect number is a number whose factors sum together to equal the
number, ex. 6 “ 1 ` 2 ` 3, or 28 “ 1 ` 2 ` 4 ` 7 ` 14.
Euclid of Alexandria, II: Number Theory
Euclid’s number theory
Definitions for number theory
Definitions for Euclidean number theory
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1: A unit is that by virtue of which each of the things that
exists is called one.
2: A number is a multitude made up of units.
5: The greater number is a multiple of the lesser when it is
measured by 3 the lesser number.
11: A prime number is that which is measured by a unit
alone.
12: Numbers prime to one another are those which are
measured by a unit alone as common measure.
13: A composite number is that which is measured by some
number.
22: A perfect number is that which is equal to its own
parts.
“Measured by” is an undefined concept in Euclid’s theory. It means
something like divided by, with no remainder.
Euclid of Alexandria, II: Number Theory
Theory of prime numbers
Properties of primes
Theory of Primes
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4
We start with some “problems” that show how to
determine if two numbers are relatively prime (VII 1), or, if
not, to find their greatest common factor (VII.2).
If p ab, then p a or p b, where p is prime (VII.30).4
“Any composite number is divisible by some prime
number.” (VII.31) [That is, given ab, there exists some p
such that p ab.] A proof by cases and by contradiction...
“If a number be the least that is measured by prime
numbers, it will not be measured by any prime number
except those originally measuring it.” (IX.14) [That is, if
a “ px11 px22 px33 ..., then there is no pn R tp1 , p2 , p3 , ...u, such
that pn a.] A proof by contradiction.
The expression a b means “a divides into b with no remainder,” or as
the Greeks would say, “a measures b.” That is, there is some n P N, such that
an “ b.
Euclid of Alexandria, II: Number Theory
Theory of prime numbers
Infinitude of primes
Elements IX.21 — The primes are infinite
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“Prime numbers are more than any assigned multitude of
prime numbers.”
A proof by construction that uses cases and an indirect
argument.
Preliminary: If g a and g b then g pa ´ bq. That is,
a “ gm and b “ gn ñ a ´ b “ gpm ´ nq.
Let tp1 , p2 , p3 u be the greatest known set of primes. Take the
number a “ p1 p2 p3 ` 1. (A construction.)
There are two cases: either, (1) a is a prime, or (2) a is
composite and has some prime factor pn .
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In case (2), we use an indirect proof to show that
pn ‰ p1 , p2 , p3 .
The arguments in Elements IX.14 and IX.21 provide a sort
of proof strategy that we could apply to any other set of
numbers that was assumed to be the set in question.
Euclid of Alexandria, II: Number Theory
Theory of prime numbers
Infinitude of primes
Euclid’s approach to studying the nature of numbers
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We begin with some “intuitive” assumptions about
numbers and some carefully designed definitions.
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We do some “problems” that show us how to carry out
certain operations or algorithms.
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We develop a theory of the concepts of measure, divisibility,
etc., which leads to a theory of primes as the fundamental
building blocks of all numbers.
§
We see again how mathematical theories help us develop
our ideas of certain key concepts.