number theory 06-08-2016corollary (parity of 0) 0 is an even number. de nition (odd numbers) an...

Number Theory

Jason Filippou

CMSC250 @ UMCP

06-08-2016

Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 1 / 1

Outline


Definition (?) of Number Theory

The Queen of Mathematics.

Study of the integers and their generalizations (primes, rationals,etc)

Used to be known as arithmetic, but nowadays arithmetic refersto first grade calculations.


Short Historical Overview




Babylonians

Figure 1: The Plimpton 322 Babylonian tablet

This tablet (created circa 1800BC) contains a series of largePythagorean triples!

Triplets of integers a, b, c such that a2 + b2 = c2



Babylonians

These guys just brute-forced those numbers



Babylonians

(3, 4, 5), (5, 12, 13)(7, 24, 25), (8, 15, 17). . . , . . .(36, 323, 325), (37, 684, 685). . . , ...

Currently believed that this plaque establishes the mathematicalidentity:

(1

2

(x− 1

x

))2+ 1 =

(1

2

(x+

1

x

))2



Egyptians, Greeks

We don’t know anything else about Babylonian Number Theory!

Babylonian algebra and astronomy, on the other hand...Also, Egyptian astronomy, geometry.Greek geometry.



Greek philosophers

The Greek mathematicians Pythagoras and Thales were influencedeither by the Babylonians or the Egyptians, or both.

Pythagorean theorem.Thales’ theorem.

Figure 2: Pythagoras of Samos. Figure 3: Thales of Miletus.



Euclid

Euclid’s Elements contain the first set of axioms of NumberTheory as we know it today.In chapters 21-34 of his 9th book of Elements, Euclid makesstatements such as:

“Odd times even is even”“If an odd number divides an even number, it also divides half of it.”

The 10th book in Elements contains a formal proof that√

2 is anirrational number.

This discovery was very upsetting for the Greeks.

Figure 4: Euclid of AlexandriaJason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 10 / 1


Chinese

The Chinese Remainder Theorem, or The Mathematical Classic ofSun TzuTM(not the famous military tactician), states:

Suppose n1, . . . , nk are integers, pairwise co-prime. Then,for any given sequence of integers a1, . . . , ak, there existsan integer x which solves the following system of equations:

x ≡ a1(mod n1)x ≡ a2(mod n2)

...

x ≡ ak(mod nk)



?



Fermat’s Last Theorem

Arguably, the most famous problem in the history of Mathematics.

Statement:

There do not exist positive integers a, b, c that satisfy theequation: an + bn = cn for values of n ≥ 3.

Fermat claimed an “elegant solution”, for which “the margin ofthe text was too small”.Finally proven by Sir Andrew Wiles, September 1994, 357 yearsafter its inception!

Figure 5: Pierre de Fermat. Figure 6: Sir Andrew Wiles.



Fermat’s Last Theorem

Arguably, the most famous problem in the history of Mathematics.Statement:

There do not exist positive integers a, b, c that satisfy theequation: an + bn = cn for values of n ≥ 3.

Fermat claimed an “elegant solution”, for which “the margin ofthe text was too small”.Finally proven by Sir Andrew Wiles, September 1994, 357 yearsafter its inception!

Figure 5: Pierre de Fermat. Figure 6: Sir Andrew Wiles.



A hard branch of Mathematics

Take-home message: Number theory is hard!

Hard to learn the math to understand it, hard to properly followthe enormous string of proofs (see: Wiles’ 1993 attempt).

In this module, we’ll attempt to give you the weaponry tomaster the latter!



Famous open problems

Hodge Conjecture.

Riemann Hypothesis.

Birch & Swinnerton-Dyer Conjecture.

Goldbach’s conjecture.

Statement:

Every even integer greater than 2 can be expressed as the sumof two primes.

Currently holds up to 4× 108, but not proven formally.


Basic Definitions

Basic Definitions


Basic Definitions

Commonly used number sets

Naturals NNaturals without zero N∗

Odd and even naturals Nodd, Neven, respectivelyIntegers ZIntegers without zero Z∗

Positive integers with zero (equiv. tonaturals)

Z+

Positive integers without zero (equiv.to N∗)

Z∗+

Negative integers with or without zero Z−, Z∗−

Odd and even integers Zodd, Zeven

Rational numbers QReal numbers RPositive, negative real numbers, with orwithout zero

R+,R−,R∗+,R∗

−

Prime numbers P

Table 1: Some commonly used number set symbols.Jason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 17 / 1

Basic Definitions

Parity

Definition (Even numbers)

An integer n is even iff a there exists an integer k such that n = 2k.

aCommon abbreviation for “if and only if”.

Corollary (Parity of 0)

0 is an even number.

Definition (Odd numbers)

An integer n is odd iff there exists an integer k such that n = 2k + 1.


Basic Definitions

Rational numbers

Definition (Rational number)

A number r is called rational iff ∃m ∈ Z, n ∈ Z∗ such that r = mn .

Corollary (Integer are rationals)

Every integer number is also rational.


Basic Definitions

Hierarchy of number sets

ℕ

ℤ

ℚ

Figure 7: Our current hierarchy of number sets.


Basic Definitions

Rational questions on rational numbers for rationalstudents

Are the following numbers rational?

1 2/32 20/303 2∗105/34 0/15 0.333333333 . . .6 0.675675675675675675 . . .7 1.435089247544 . . .8 π, φ, e,

√2

So there’s something beyond rational numbers!


Basic Definitions


Are the following numbers rational?1 2/3

2 20/303 2∗105/34 0/15 0.333333333 . . .6 0.675675675675675675 . . .7 1.435089247544 . . .8 π, φ, e,

√2



Basic Definitions


Are the following numbers rational?1 2/32 20/30

3 2∗105/34 0/15 0.333333333 . . .6 0.675675675675675675 . . .7 1.435089247544 . . .8 π, φ, e,

√2



Basic Definitions


Are the following numbers rational?1 2/32 20/303 2∗105/3

4 0/15 0.333333333 . . .6 0.675675675675675675 . . .7 1.435089247544 . . .8 π, φ, e,

√2



Basic Definitions


Are the following numbers rational?1 2/32 20/303 2∗105/34 0/1

5 0.333333333 . . .6 0.675675675675675675 . . .7 1.435089247544 . . .8 π, φ, e,

√2



Basic Definitions


Are the following numbers rational?1 2/32 20/303 2∗105/34 0/15 0.333333333 . . .

6 0.675675675675675675 . . .7 1.435089247544 . . .8 π, φ, e,

√2



Basic Definitions


Are the following numbers rational?1 2/32 20/303 2∗105/34 0/15 0.333333333 . . .6 0.675675675675675675 . . .

7 1.435089247544 . . .8 π, φ, e,

√2



Basic Definitions


Are the following numbers rational?1 2/32 20/303 2∗105/34 0/15 0.333333333 . . .6 0.675675675675675675 . . .7 1.435089247544 . . .

8 π, φ, e,√

2



Basic Definitions


Are the following numbers rational?1 2/32 20/303 2∗105/34 0/15 0.333333333 . . .6 0.675675675675675675 . . .7 1.435089247544 . . .8 π, φ, e,

√2



Basic Definitions

Irrational numbers

Definition (Irrational numbers)

A number is irrational iff it is not rational.

Corollary

If s is an irrational number, there does not exist a pair of integers m,n, with n 6= 0, such that s = m

n .

Rationals and irrationals together give us the set of real numbers:R.


Basic Definitions

Hierarchy of number sets, revisited

N

Z

Q

R–Q

R

Figure 8: Our hierarchy, updated.

We will not deal with higher number systems (complex numbers,quaternions).Note that rationals and irrationals complement each other(obviously).


Basic Definitions

Prime numbers

Definition (Prime number)

An integer n ≥ 2 is called prime iff its only factors (divisors) are 1 andn.

Corollary (Primality of 2)

2 is the only even prime number.

Prime numbers are fundamental in Number Theory, for a varietyof reasons.

Important enough that the set of primes has a symbol: P

Largest known prime: 274,207,281 − 1 (22,338,618 digits).

Discovered 01-2016 by the Great Internet Mersenne Prime Search(GIMPS).


Basic Definitions

Composite numbers

Definition (Composite number)

An integer n ≥ 2 is called composite iff it is not prime.

Corollary (Primality of 0 and 1)

1 and 0 are neither prime nor composite.


Basic Definitions

Hierarchy of number sets, revisited

N

Z

Q

R–Q

R

P

Figure 9: Our final hierarchy.

Possible to define more sets, like even and odd integers, Mersenneprimes, etc.


number theory 06-08-2016corollary (parity of 0) 0 is an even number. de nition (odd numbers) an...

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