Math 312, Lecture 1reichst/312lecture1.pdf · Number theory ourished in ancient Greece ... number...

Math 312, Lecture 1

Zinovy Reichstein

September 9, 2015Math 312

Math 312, Lecture 1 September 9, 2015

Number theory

Number theory is a branch of mathematics

which studies the properties ofintegers (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 callPythagorean triples, (a, b, c) of positive integers such that a2 + b2 = c2.

Number theory flourished in ancient Greece (as did other areas ofmathematics). Some of the foundaional concepts still bear the name ofArchimedes, Diophantus, Euclid, Pythagoras.

The discoveries of ancient greek mathematicians were “lost” in Europeduring the dark ages, over a period of several centuries. During this periodnumber theory continued to be advanced in China, India and the MiddleEast. The concept of zero and positional representation of numbers wereknown to Indian mathematicians by the 7th century. Important advancesin algebra were made by Persian and Arab mathematicians throughout thisperiod.


Number theory

Number theory is a branch of mathematics which studies the properties ofintegers (or whole numbers).

Its origins go back to (at least) 4000 years,to ancient Baylon.





Number theory

Number theory is a branch of mathematics which studies the properties ofintegers (or whole numbers). Its origins go back to (at least) 4000 years,to ancient Baylon.





Number theory



Number theory flourished in ancient Greece (as did other areas ofmathematics).

Some of the foundaional concepts still bear the name ofArchimedes, Diophantus, Euclid, Pythagoras.



Number theory






Number theory




The discoveries of ancient greek mathematicians were “lost” in Europeduring the dark ages, over a period of several centuries.

During this periodnumber theory continued to be advanced in China, India and the MiddleEast. The concept of zero and positional representation of numbers wereknown to Indian mathematicians by the 7th century. Important advancesin algebra were made by Persian and Arab mathematicians throughout thisperiod.


Number theory






Fermat andd Euler

During the Renaissance classical number theory returned to Europe. Mostof the theoretical material in this course is based on the work of greatnumber theorists of the 17th and the 18th centuries, Pierre de Fermat(1601-1665) and Leonhard Euler (1707-1783).


Fermat andd Euler

During the Renaissance classical number theory returned to Europe.

Mostof the theoretical material in this course is based on the work of greatnumber theorists of the 17th and the 18th centuries, Pierre de Fermat(1601-1665) and Leonhard Euler (1707-1783).


Fermat andd Euler

During the Renaissance classical number theory returned to Europe. Mostof the theoretical material in this course is based on the work of greatnumber theorists of the 17th and the 18th centuries,

Pierre de Fermat(1601-1665) and Leonhard Euler (1707-1783).


Fermat andd Euler

During the Renaissance classical number theory returned to Europe. Mostof the theoretical material in this course is based on the work of greatnumber theorists of the 17th and the 18th centuries, Pierre de Fermat(1601-1665) and Leonhard Euler (1707-1783).


Pierre de Fermat (1601-1665), first photo

Fermat was a prominent Frenchlawer, whose work combined classicalnumber theory with newly developedalgebraic methods.


Pierre de Fermat (1601-1665), second photo

Fermat’s “Last Theorem”: Let n > 3be an integer. Then xn + yn 6= zn forany triple of positive integers x , y , z .


Leonhard Euler (1707-1783)

Leonhard Euler isconsidered to be thepreeminent 18th centurymathematician and oneof the greatestmathematicians inhistory. He worked in arange of subjects,including number theory,graph theory andcalculus.


Leonhard Euler (1707-1783), second photo

Leonhard Euler was Swiss, but hespent most of his life in Germany(Prussia) and Russia (St.Petersburg). In addition inmathematics, he was also renownedfor his work in mechanics, fluiddynamics, optics, astronomy, andmusic theory.



Leonhard Euler was Swiss,

but hespent most of his life in Germany(Prussia) and Russia (St.Petersburg). In addition inmathematics, he was also renownedfor his work in mechanics, fluiddynamics, optics, astronomy, andmusic theory.



Leonhard Euler was Swiss, but hespent most of his life in Germany(Prussia) and Russia (St.Petersburg).

In addition inmathematics, he was also renownedfor his work in mechanics, fluiddynamics, optics, astronomy, andmusic theory.



Leonhard Euler was Swiss, but hespent most of his life in Germany(Prussia) and Russia (St.Petersburg). In addition inmathematics, he was also renowned

for his work in mechanics, fluiddynamics, optics, astronomy, andmusic theory.



Leonhard Euler was Swiss, but hespent most of his life in Germany(Prussia) and Russia (St.Petersburg). In addition inmathematics, he was also renownedfor his work in mechanics, fluiddynamics, optics, astronomy, andmusic theory.


Applications of number theory

Number theory was long viewed as “The queen of mathematics”,

an areaof pure research, akin to painting or poetry, valued for its intrinsic beauty,not applicability.

This view was most directly expressed in the 1940 essay “Amathematician’s apology” by the British mathematician G.H. Hardy.

“No one has yet discovered any warlike purpose to be served by the theoryof numbers or relativity, and it seems unlikely that anyone will do so formany years.”

This turned out to be spectacularly false. Number theory turned out tohave far-reaching applications in the computer age, both industrial andmilitary.

Applications, to cryptography, will be covered in the course.



Number theory was long viewed as “The queen of mathematics”, an areaof pure research, akin to painting or poetry,

valued for its intrinsic beauty,not applicability.







Number theory was long viewed as “The queen of mathematics”, an areaof pure research, akin to painting or poetry, valued for its intrinsic beauty,not applicability.








This view was most directly expressed in the 1940 essay “Amathematician’s apology”

by the British mathematician G.H. Hardy.








“No one has yet discovered any warlike purpose to be served by the theoryof numbers or relativity,

and it seems unlikely that anyone will do so formany years.”




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.


Mathematical induction

Suppose a set S of positive integers (i) contains 1, and (ii) has theproperty that if S contains n then S contains n + 1.

Then S containsevery positive integer, i.e., S = N.

In practice we use this to prove that some property Pn is satisfied by everypositive 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 iscalled “the induction step”.

If we can establish (i) and (ii), then property Pn will be true for everyinteger n > 1. To see that this proof method is valid, denote the set ofintegers n such that Pn is satisfied by S , and use the principle ofmathematical induction to show that S = N.


Mathematical induction

Suppose a set S of positive integers (i) contains 1, and (ii) has theproperty that if S contains n then S contains n + 1. Then S containsevery positive integer, i.e., S = N.


(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 iscalled “the induction step”.

If we can establish (i) and (ii), then property Pn will be true for everyinteger n > 1. To see that this proof method is valid, denote the set ofintegers n such that Pn is satisfied by S , and use the principle ofmathematical induction to show that S = N.


Mathematical induction examples/exercises

1. Show that 1 + 3 + 5 + · · ·+ (2n − 1) = n2.

2. Show that 1 + q + q2 + · · ·+ qn =qn+1 − 1

q − 1for any real number q 6= 1.

3. Show that 2n > n2 for any n > 5.

4. Show that n lines in general position subdivide the plane inton(n + 1)

2+ 1 regions.

5. Show that n3 − n is divisible by 3 for any n > 1.


Strong mathematical induction

The following variant on the principle of mathematical induction is oftenconvenient.

Suppose a set S of positive integers (i) contains 1, and (ii) has theproperty that if S contains every integer 6 n then S contains n + 1.Then S contains every positive integer, i.e., S = N.


Base case: First we prove that P1 is satisfied.

Induction step: Then we prove that if P1, . . . ,Pn are all satisfied, thenPn+1 is satisfied as well.




Suppose a set S of positive integers (i) contains 1, and (ii) has theproperty that if S contains every integer 6 n then S contains n + 1.

Then S contains every positive integer, i.e., S = N.







Suppose a set S of positive integers (i) contains 1, and (ii) has theproperty that if S contains every integer 6 n then S contains n + 1.Then S contains every positive integer, i.e., S = N.





Strong mathematical induction exercises

1. Show that every integer n > 2 is either a prime or a product of two ormore primes.

2. Any integer amount of postage of 12 cents or more, can be paid usingonly 3-cent and 5-cent stamps.

3. The nth Fibonacci number an is defined by the recursive formulaa1 = a2 = 1, an+2 = an+1 + an. Show that

an =1√5

(αn − βn)

for any n > 1. Here

α =1 +√

5

2and β =

1−√

5

2.

Note that α and β are the roots of the quadratic equation x2 − x − 1 = 0.That is, α2 = α + 1 and β2 = β + 1.






an =1√5

(αn − βn)

for any n > 1. Here

α =1 +√

5

2and β =

1−√

5

2.

Note that α and β are the roots of the quadratic equation x2 − x − 1 = 0.

That is, α2 = α + 1 and β2 = β + 1.






an =1√5

(αn − βn)

for any n > 1. Here

α =1 +√

5

2and β =

1−√

5

2.

Note that α and β are the roots of the quadratic equation x2 − x − 1 = 0.That is, α2 = α + 1 and β2 = β + 1.


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 withrespect to m. Note that m > 1 corresponds to n > d .

Remark 2. The well-ordering principle, the principle of mathematicalinduction and the principle of strong mathematical induction are allequivalent to each other.


Remarks





Remark 2. The well-ordering principle, the principle of mathematicalinduction

and the principle of strong mathematical induction are allequivalent to each other.


Remarks





Remark 2. The well-ordering principle, the principle of mathematicalinduction and the principle of strong mathematical induction

are allequivalent to each other.


Remarks





Remark 2. The well-ordering principle, the principle of mathematicalinduction and the principle of strong mathematical induction are allequivalent to each other.


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