Math 312, Lecture 1
Zinovy Reichstein
September 9, 2015
Math 312
Math 312, Lecture 1
September 9, 2015
Number theory
Number theory is a branch of mathematics
Math 312, Lecture 1
September 9, 2015
Number theory
Number theory is a branch of mathematics which studies the properties of
integers (or whole numbers).
Math 312, Lecture 1
September 9, 2015
Number theory
Number theory is a branch of mathematics which studies the properties of
integers (or whole numbers). Its origins go back to (at least) 4000 years,
to ancient Baylon.
Math 312, Lecture 1
September 9, 2015
Number theory
Number theory is a branch of mathematics which studies the properties of
integers (or whole numbers). Its origins go back to (at least) 4000 years,
to ancient Baylon.
In particular, the ancient Babylonians knew of what we now call
Pythagorean triples, (a, b, c) of positive integers such that a2 + b 2 = c 2 .
Math 312, Lecture 1
September 9, 2015
Number theory
Number theory is a branch of mathematics which studies the properties of
integers (or whole numbers). Its origins go back to (at least) 4000 years,
to ancient Baylon.
In particular, the ancient Babylonians knew of what we now call
Pythagorean triples, (a, b, c) of positive integers such that a2 + b 2 = c 2 .
Number theory flourished in ancient Greece (as did other areas of
mathematics).
Math 312, Lecture 1
September 9, 2015
Number theory
Number theory is a branch of mathematics which studies the properties of
integers (or whole numbers). Its origins go back to (at least) 4000 years,
to ancient Baylon.
In particular, the ancient Babylonians knew of what we now call
Pythagorean triples, (a, b, c) of positive integers such that a2 + b 2 = c 2 .
Number theory flourished in ancient Greece (as did other areas of
mathematics). Some of the foundaional concepts still bear the name of
Archimedes, Diophantus, Euclid, Pythagoras.
Math 312, Lecture 1
September 9, 2015
Number theory
Number theory is a branch of mathematics which studies the properties of
integers (or whole numbers). Its origins go back to (at least) 4000 years,
to ancient Baylon.
In particular, the ancient Babylonians knew of what we now call
Pythagorean triples, (a, b, c) of positive integers such that a2 + b 2 = c 2 .
Number theory flourished in ancient Greece (as did other areas of
mathematics). Some of the foundaional concepts still bear the name of
Archimedes, Diophantus, Euclid, Pythagoras.
The discoveries of ancient greek mathematicians were “lost” in Europe
during the dark ages, over a period of several centuries.
Math 312, Lecture 1
September 9, 2015
Number theory
Number theory is a branch of mathematics which studies the properties of
integers (or whole numbers). Its origins go back to (at least) 4000 years,
to ancient Baylon.
In particular, the ancient Babylonians knew of what we now call
Pythagorean triples, (a, b, c) of positive integers such that a2 + b 2 = c 2 .
Number theory flourished in ancient Greece (as did other areas of
mathematics). Some of the foundaional concepts still bear the name of
Archimedes, Diophantus, Euclid, Pythagoras.
The discoveries of ancient greek mathematicians were “lost” in Europe
during the dark ages, over a period of several centuries. During this period
number theory continued to be advanced in China, India and the Middle
East. The concept of zero and positional representation of numbers were
known to Indian mathematicians by the 7th century. Important advances
in algebra were made by Persian and Arab mathematicians throughout this
period.
Math 312, Lecture 1
September 9, 2015
Fermat andd Euler
Math 312, Lecture 1
September 9, 2015
Fermat andd Euler
During the Renaissance classical number theory returned to Europe.
Math 312, Lecture 1
September 9, 2015
Fermat andd Euler
During the Renaissance classical number theory returned to Europe. Most
of the theoretical material in this course is based on the work of great
number theorists of the 17th and the 18th centuries,
Math 312, Lecture 1
September 9, 2015
Fermat andd Euler
During the Renaissance classical number theory returned to Europe. Most
of the theoretical material in this course is based on the work of great
number theorists of the 17th and the 18th centuries, Pierre de Fermat
(1601-1665) and Leonhard Euler (1707-1783).
Math 312, Lecture 1
September 9, 2015
Pierre de Fermat (1601-1665), first photo
Math 312, Lecture 1
September 9, 2015
Pierre de Fermat (1601-1665), first photo
Fermat was a prominent French
lawer, whose work combined classical
number theory with newly developed
algebraic methods.
Math 312, Lecture 1
September 9, 2015
Pierre de Fermat (1601-1665), second photo
Math 312, Lecture 1
September 9, 2015
Pierre de Fermat (1601-1665), second photo
Fermat’s “Last Theorem”: Let n > 3
be an integer. Then x n + y n 6= z n for
any triple of positive integers x, y , z.
Math 312, Lecture 1
September 9, 2015
Leonhard Euler (1707-1783)
Math 312, Lecture 1
September 9, 2015
Leonhard Euler (1707-1783)
Leonhard Euler is
considered to be the
preeminent 18th century
mathematician and one
of the greatest
mathematicians in
history. He worked in a
range of subjects,
including number theory,
graph theory and
calculus.
Math 312, Lecture 1
September 9, 2015
Leonhard Euler (1707-1783), second photo
Math 312, Lecture 1
September 9, 2015
Leonhard Euler (1707-1783), second photo
Leonhard Euler was Swiss,
Math 312, Lecture 1
September 9, 2015
Leonhard Euler (1707-1783), second photo
Leonhard Euler was Swiss, but he
spent most of his life in Germany
(Prussia) and Russia (St.
Petersburg).
Math 312, Lecture 1
September 9, 2015
Leonhard Euler (1707-1783), second photo
Leonhard Euler was Swiss, but he
spent most of his life in Germany
(Prussia) and Russia (St.
Petersburg). In addition in
mathematics, he was also renowned
Math 312, Lecture 1
September 9, 2015
Leonhard Euler (1707-1783), second photo
Leonhard Euler was Swiss, but he
spent most of his life in Germany
(Prussia) and Russia (St.
Petersburg). In addition in
mathematics, he was also renowned
for his work in mechanics, fluid
dynamics, optics, astronomy, and
music theory.
Math 312, Lecture 1
September 9, 2015
Applications of number theory
Number theory was long viewed as “The queen of mathematics”,
Math 312, Lecture 1
September 9, 2015
Applications of number theory
Number theory was long viewed as “The queen of mathematics”, an area
of pure research, akin to painting or poetry,
Math 312, Lecture 1
September 9, 2015
Applications of number theory
Number theory was long viewed as “The queen of mathematics”, an area
of pure research, akin to painting or poetry, valued for its intrinsic beauty,
not applicability.
Math 312, Lecture 1
September 9, 2015
Applications of number theory
Number theory was long viewed as “The queen of mathematics”, an area
of pure research, akin to painting or poetry, valued for its intrinsic beauty,
not applicability.
This view was most directly expressed in the 1940 essay “A
mathematician’s apology”
Math 312, Lecture 1
September 9, 2015
Applications of number theory
Number theory was long viewed as “The queen of mathematics”, an area
of pure research, akin to painting or poetry, valued for its intrinsic beauty,
not applicability.
This view was most directly expressed in the 1940 essay “A
mathematician’s apology” by the British mathematician G.H. Hardy.
Math 312, Lecture 1
September 9, 2015
Applications of number theory
Number theory was long viewed as “The queen of mathematics”, an area
of pure research, akin to painting or poetry, valued for its intrinsic beauty,
not applicability.
This view was most directly expressed in the 1940 essay “A
mathematician’s apology” by the British mathematician G.H. Hardy.
“No one has yet discovered any warlike purpose to be served by the theory
of numbers or relativity,
Math 312, Lecture 1
September 9, 2015
Applications of number theory
Number theory was long viewed as “The queen of mathematics”, an area
of pure research, akin to painting or poetry, valued for its intrinsic beauty,
not applicability.
This view was most directly expressed in the 1940 essay “A
mathematician’s apology” by the British mathematician G.H. Hardy.
“No one has yet discovered any warlike purpose to be served by the theory
of numbers or relativity, and it seems unlikely that anyone will do so for
many years.”
Math 312, Lecture 1
September 9, 2015
Applications of number theory
Number theory was long viewed as “The queen of mathematics”, an area
of pure research, akin to painting or poetry, valued for its intrinsic beauty,
not applicability.
This view was most directly expressed in the 1940 essay “A
mathematician’s apology” by the British mathematician G.H. Hardy.
“No one has yet discovered any warlike purpose to be served by the theory
of numbers or relativity, and it seems unlikely that anyone will do so for
many years.”
This turned out to be spectacularly false. Number theory turned out to
have far-reaching applications in the computer age, both industrial and
military.
Math 312, Lecture 1
September 9, 2015
Applications of number theory
Number theory was long viewed as “The queen of mathematics”, an area
of pure research, akin to painting or poetry, valued for its intrinsic beauty,
not applicability.
This view was most directly expressed in the 1940 essay “A
mathematician’s apology” by the British mathematician G.H. Hardy.
“No one has yet discovered any warlike purpose to be served by the theory
of numbers or relativity, and it seems unlikely that anyone will do so for
many years.”
This turned out to be spectacularly false. Number theory turned out to
have far-reaching applications in the computer age, both industrial and
military.
Applications, to cryptography, will be covered in the course.
Math 312, Lecture 1
September 9, 2015
The Well-Ordering Principle
Every subset of the positive integers has a least (i.e., the smallest)
element.
Math 312, Lecture 1
September 9, 2015
The Well-Ordering Principle
Every subset of the positive integers has a least (i.e., the smallest)
element.
For us this will be one of the axioms of the natural numbers.
Math 312, Lecture 1
September 9, 2015
The Well-Ordering Principle
Every subset of the positive integers has a least (i.e., the smallest)
element.
For us this will be one of the axioms of the natural numbers.
Note that the well-ordering principle is false for other number systems,
such as all integers or the positive real numbers.
Math 312, Lecture 1
September 9, 2015
Mathematical induction
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains n then S contains n + 1.
Math 312, Lecture 1
September 9, 2015
Mathematical induction
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains n then S contains n + 1. Then S contains
every positive integer, i.e., S = N.
Math 312, Lecture 1
September 9, 2015
Mathematical induction
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains n then S contains n + 1. Then S contains
every positive integer, i.e., S = N.
In practice we use this to prove that some property Pn is satisfied by every
positive integer n as follows.
Math 312, Lecture 1
September 9, 2015
Mathematical induction
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains n then S contains n + 1. Then S contains
every positive integer, i.e., S = N.
In practice we use this to prove that some property Pn is satisfied by every
positive integer n as follows.
(i) First we prove that P1 is satisfied. This is called “the base case”.
Math 312, Lecture 1
September 9, 2015
Mathematical induction
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains n then S contains n + 1. Then S contains
every positive integer, i.e., S = N.
In practice we use this to prove that some property Pn is satisfied by every
positive integer n as follows.
(i) First we prove that P1 is satisfied. This is called “the base case”.
(ii) Then we prove that if Pn is satisfied, then Pn+1 is satisfied. This is
called “the induction step”.
Math 312, Lecture 1
September 9, 2015
Mathematical induction
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains n then S contains n + 1. Then S contains
every positive integer, i.e., S = N.
In practice we use this to prove that some property Pn is satisfied by every
positive integer n as follows.
(i) First we prove that P1 is satisfied. This is called “the base case”.
(ii) Then we prove that if Pn is satisfied, then Pn+1 is satisfied. This is
called “the induction step”.
If we can establish (i) and (ii), then property Pn will be true for every
integer n > 1. To see that this proof method is valid, denote the set of
integers n such that Pn is satisfied by S, and use the principle of
mathematical induction to show that S = N.
Math 312, Lecture 1
September 9, 2015
Mathematical induction examples/exercises
1. Show that 1 + 3 + 5 + · · · + (2n − 1) = n2 .
2. Show that 1 + q + q 2 + · · · + q n =
q n+1 − 1
for any real number q 6= 1.
q−1
3. Show that 2n > n2 for any n > 5.
4. Show that n lines in general position subdivide the plane into
n(n + 1)
+ 1 regions.
2
5. Show that n3 − n is divisible by 3 for any n > 1.
Math 312, Lecture 1
September 9, 2015
Strong mathematical induction
The following variant on the principle of mathematical induction is often
convenient.
Math 312, Lecture 1
September 9, 2015
Strong mathematical induction
The following variant on the principle of mathematical induction is often
convenient.
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains every integer 6 n then S contains n + 1.
Math 312, Lecture 1
September 9, 2015
Strong mathematical induction
The following variant on the principle of mathematical induction is often
convenient.
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains every integer 6 n then S contains n + 1.
Then S contains every positive integer, i.e., S = N.
Math 312, Lecture 1
September 9, 2015
Strong mathematical induction
The following variant on the principle of mathematical induction is often
convenient.
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains every integer 6 n then S contains n + 1.
Then S contains every positive integer, i.e., S = N.
In practice we use this to prove that some property Pn is satisfied by every
positive integer n as follows.
Math 312, Lecture 1
September 9, 2015
Strong mathematical induction
The following variant on the principle of mathematical induction is often
convenient.
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains every integer 6 n then S contains n + 1.
Then S contains every positive integer, i.e., S = N.
In practice we use this to prove that some property Pn is satisfied by every
positive integer n as follows.
Base case: First we prove that P1 is satisfied.
Math 312, Lecture 1
September 9, 2015
Strong mathematical induction
The following variant on the principle of mathematical induction is often
convenient.
Suppose a set S of positive integers (i) contains 1, and (ii) has the
property that if S contains every integer 6 n then S contains n + 1.
Then S contains every positive integer, i.e., S = N.
In practice we use this to prove that some property Pn is satisfied by every
positive integer n as follows.
Base case: First we prove that P1 is satisfied.
Induction step: Then we prove that if P1 , . . . , Pn are all satisfied, then
Pn+1 is satisfied as well.
Math 312, Lecture 1
September 9, 2015
Strong mathematical induction exercises
1. Show that every integer n > 2 is either a prime or a product of two or
more primes.
2. Any integer amount of postage of 12 cents or more, can be paid using
only 3-cent and 5-cent stamps.
3. The nth Fibonacci number an is defined by the recursive formula
a1 = a2 = 1, an+2 = an+1 + an . Show that
1
an = √ (αn − β n )
5
for any n > 1. Here
√
√
1+ 5
1− 5
α=
and β =
.
2
2
Math 312, Lecture 1
September 9, 2015
Strong mathematical induction exercises
1. Show that every integer n > 2 is either a prime or a product of two or
more primes.
2. Any integer amount of postage of 12 cents or more, can be paid using
only 3-cent and 5-cent stamps.
3. The nth Fibonacci number an is defined by the recursive formula
a1 = a2 = 1, an+2 = an+1 + an . Show that
1
an = √ (αn − β n )
5
for any n > 1. Here
√
√
1+ 5
1− 5
α=
and β =
.
2
2
Note that α and β are the roots of the quadratic equation x 2 − x − 1 = 0.
Math 312, Lecture 1
September 9, 2015
Strong mathematical induction exercises
1. Show that every integer n > 2 is either a prime or a product of two or
more primes.
2. Any integer amount of postage of 12 cents or more, can be paid using
only 3-cent and 5-cent stamps.
3. The nth Fibonacci number an is defined by the recursive formula
a1 = a2 = 1, an+2 = an+1 + an . Show that
1
an = √ (αn − β n )
5
for any n > 1. Here
√
√
1+ 5
1− 5
α=
and β =
.
2
2
Note that α and β are the roots of the quadratic equation x 2 − x − 1 = 0.
That is, α2 = α + 1 and β 2 = β + 1.
Math 312, Lecture 1
September 9, 2015
Remarks
Remark 1. Mathematical induction can be used to prove that property Pn
Math 312, Lecture 1
September 9, 2015
Remarks
Remark 1. Mathematical induction can be used to prove that property Pn
is valid for every integer n > d, where d is not necessarily 1.
Math 312, Lecture 1
September 9, 2015
Remarks
Remark 1. Mathematical induction can be used to prove that property Pn
is valid for every integer n > d, where d is not necessarily 1.
For example, 2n > n2 for every n > 5.
Math 312, Lecture 1
September 9, 2015
Remarks
Remark 1. Mathematical induction can be used to prove that property Pn
is valid for every integer n > d, where d is not necessarily 1.
For example, 2n > n2 for every n > 5.
Here we use the substitution m = n − d + 1 and argue by induction with
respect to m. Note that m > 1 corresponds to n > d.
Remark 2. The well-ordering principle, the principle of mathematical
induction
Math 312, Lecture 1
September 9, 2015
Remarks
Remark 1. Mathematical induction can be used to prove that property Pn
is valid for every integer n > d, where d is not necessarily 1.
For example, 2n > n2 for every n > 5.
Here we use the substitution m = n − d + 1 and argue by induction with
respect to m. Note that m > 1 corresponds to n > d.
Remark 2. The well-ordering principle, the principle of mathematical
induction and the principle of strong mathematical induction
Math 312, Lecture 1
September 9, 2015
Remarks
Remark 1. Mathematical induction can be used to prove that property Pn
is valid for every integer n > d, where d is not necessarily 1.
For example, 2n > n2 for every n > 5.
Here we use the substitution m = n − d + 1 and argue by induction with
respect to m. Note that m > 1 corresponds to n > d.
Remark 2. The well-ordering principle, the principle of mathematical
induction and the principle of strong mathematical induction are all
equivalent to each other.
Math 312, Lecture 1
September 9, 2015