Prime Numbers and the Convergents of a Continued Fraction

7/30/2019 Prime Numbers and the Convergents of a Continued Fraction

1/58

Prime Numbers and the Convergents of a ContinuedFraction

Cahlen Humphreys

Boise State UniversityIndependent Study
http://find/http://goback/


2/58

Introduction

Figure: John Wallis 1616-1703

The term Continued Fraction was first coined by John Wallis in his bookOpera Mathematica in 1693. In this book he describes what continuedfractions are and what the nth convergent of a continued fraction is, as

well as properties of convergents that we use today.C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction

2 / 30
http://find/


3/58

Introduction

Continued fractions have applications in

C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction

3 / 30
http://find/


4/58

Introduction


1 Mathematical Cryptography [1].

C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 3 / 30
http://find/


5/58

Introduction



2 Atomic Physics [2].

http://find/


6/58


7/58

Introduction




3 Cosmology [3].

4 Ecology [4].

http://find/


8/58

Introduction




3 Cosmology [3].

4 Ecology [4].

The security of many cryptosystems rely on a computationally difficultmathematical problem (see eg. [1], [5], [6]).

http://find/


9/58

Introduction

The security of one such cryptosystem in particular called RSA relies onthe difficulty of a factoring problem [1]. Factoring algorithms and attackshave been developed using continued fractions that help defeat thesecurity of this cryptosystem.



10/58

Introduction


1 Wieners Method [7].

http://find/


11/58

Introduction


1 Wieners Method [7].

2 CFRAC [8].



12/58

Introduction

A continued fraction of the form

a0 +b0

a1 +

b1

a2 + . . .

where a0, a1, a2, . . . and b0, b1, . . . may be real or complex numbers. ASimple Continued Fraction is of the same form exceptb0 = b1 = b2 =

= 1, a0 is either positive, negative, or zero, and

a1, a2, . . . are all positive integers.



13/58

Introduction

We use the alternate notation

a0 + 1a1+ 1

a2+ 1

a3+

and

[a0, a1, a2, a3, . . . ]

instead of the traditional notation for the expansion of a simple continuedfraction, primarily to save space.



14/58

Introduction

We use the alternate notation

a0 + 1a1+ 1

a2+ 1

a3+

and

[a0, a1, a2, a3, . . . ]

instead of the traditional notation for the expansion of a simple continuedfraction, primarily to save space.

Theorem 1 (Euler, [11])

A number is rational if and only if it can expressed as a simple finitecontinued fraction. Similarly, a number is irrational if and only if it can beexpressed as an simple infinite continued fraction.



15/58

Convergents

Definition 2 (Wallis, [9])

The continued fraction [a1, a2, a3, . . . , an] where n is a non-negative integerless than or equal to k is called the nth convergent of the continuedfraction [a1, a2, a3, . . . , ak]. The n

th convergent is denoted by Cn.

http://find/


16/58

Convergents

Definition 2 (Wallis, [9])

The continued fraction [a1, a2, a3, . . . , an] where n is a non-negative integerless than or equal to k is called the nth convergent of the continuedfraction [a1, a2, a3, . . . , ak]. The n

th convergent is denoted by Cn.

Convergents of a continued fraction are essentially a segment of a

continued fraction that we compute. For example, let

= a0 +1

a1+

1

a2+ + 1

ak

then a convergent Cn

of for n < k is

Cn =AnBn

= a0 +1

a1+

1

a2+ + 1

an Finite, by Theorem 1 a rational number.

.


C
http://find/


17/58

Convergents

We also have some very nice properties that involve the convergents of acontinued fraction.


C
http://find/


18/58

Convergents

We also have some very nice properties that involve the convergents of acontinued fraction.

Theorem 3 (Olds, [10])

The numerators An and the denominators Bn of the nth convergent Cn of

the continued fraction

a1 + 1a2+ 1

a3+ + 1

ak

satisfy the equations

An = anAn1 + An2,Bn = anBn1 + Bn2,

where (n = 3, 4, 5, . . . , k), with the initial valuesA1 = a1, A2 = a2a1 + 1, B1 = 1, and B2 = a2.


C
http://find/


19/58

Convergents

Theorem 4 (Dirichlet, [12])Let x be irrational, then there are infinitely many rationals numbers A

B

such that

xA

B 1

B2 .


C t
http://find/


20/58

Convergents


B

such that

xA

B 1

B2 .

Manufacturers will sometimes create cryptosystems with backdoors as anyeasy way to gain access in the event that a private key is lost.


C t
http://find/


21/58

Convergents


B

such that

xA

B 1

B2 .

Manufacturers will sometimes create cryptosystems with backdoors as anyeasy way to gain access in the event that a private key is lost.Andersons RSA Trapdoor [13] is one such backdoor. Andersons RSA

Trapdoor generates primes p and p where N = pp is easy to factor by themanufacturer. Kaliski proved it could be broken with help from Theorem 4[14].


Convergents
http://find/


22/58

Convergents

Theorem 5 (Dirichlet, [12])

Let x be irrational, and let AB

be a rational in lowest terms with B > 0,suppose that

x AB 12B2 .

Then AB

is a convergent in the continued fraction expansion of x.


Big O Notation
http://find/


23/58

Big O Notation

Big O Notation, also called Landaus Symbol, is a symbolism used inmathematics and computer science that is used to describe the asymptotic

nature of functions.


Big O Notation
http://find/


24/58

Big O Notation


nature of functions.Definition 6 (Bachmann, [16])

Suppose f(x) and g(x) are two functions defined on a subset of the realnumbers, we write f(x) = O (g(x)) if an only if there exists constants Nand C such that |f(x)| C|g(x)| for all x > N.


Big O Notation
http://find/


25/58

Big O Notation



Suppose f(x) and g(x) are two functions defined on a subset of the realnumbers, we write f(x) = O (g(x)) if an only if there exists constants Nand C such that |f(x)| C|g(x)| for all x > N.Theorem 7 (Bachmann, [16])

If f1(x) = O(g1(x)) and f2(x) = O(g2(x)), thenf

1(x) + f

2(x) =

O(max (g

1(x), g

2(x))).


Big O Notation
http://find/


26/58

Big O Notation



Suppose f(x) and g(x) are two functions defined on a subset of the realnumbers, we write f(x) = O (g(x)) if an only if there exists constants Nand C such that |f(x)| C|g(x)| for all x > N.Theorem 7 (Bachmann, [16])

If f1(x) = O(g1(x)) and f2(x) = O(g2(x)), thenf

1(x) + f

2(x) =

O(max (g

1(x), g

2(x))).

Theorem 8 (Bachmann, [16])

If f1(x) = O(g1(x)) and f2(x) = O(g2(x)), thenf1(x)

f2(x) =

O(g1(x)

g2(x)).


Motivation
http://find/


27/58

Motivation


Motivation
http://find/


28/58

Motivation

In 1939 P. Erdos and K. Mahler showed that there are irrational numbersfor which each of the denominators of the convergents of their continuedfraction expansion is a prime number.


Motivation
http://find/


29/58

Motivation

In 1939 P. Erdos and K. Mahler showed that there are irrational numbersfor which each of the denominators of the convergents of their continuedfraction expansion is a prime number.

Theorem 9 (P. Erdos, K. Mahler, [15])

For almost all irrational numbers , the greatest prime factor of thedenominator Bn of the n

th convergent Cn = An/Bn of the continuedfraction expansion of , increases rapidly with n.


Our Research
http://find/


30/58

Our Research

In our research we focus on the numerator An instead of Bn.


Our Research
http://find/


31/58

Our Research

In our research we focus on the numerator An instead of Bn.

Theorem 10

For almost all irrational numbers , the greatest prime factor of thenumerator An of the n

th convergent Cn = An/Bn of the continued fractionexpansion of , increases rapidly with n.


Our Research
http://find/


32/58

Denote G(k) as the greatest prime factor of the number k.


Our Research
http://find/


33/58

Denote G(k) as the greatest prime factor of the number k.

Theorem 11 (P. Erdos, K. Mahler, [15], Lemma 1)

Let S be the set of all positive integers k for which

k , G(k) e lnk

20 ln (ln k)

then, for large > 0,

kS

1

k=

O (ln )3 .


Our Research


34/58

We give a much more detailed description of Theorem 11 by splitting it up

into three Lemmas.


Our Research
http://find/


35/58


into three Lemmas. Let L = e

ln k20 ln ln k

and S = {k : G(k) L}, andS = Ti where T = {k x : G(k) L}.Lemma 12

If A =

{k

Ti : (

r

N)(r2

(ln x)10 and r2

|k)

}, then

|A

|=

O

x

(ln x)5

.


Our Research
http://find/


36/58



ln k20 ln ln k


If A =

{k

Ti : (

r

N)(r2

(ln x)10 and r2

|k)

}, then

|A

|=

O

x

(ln x)5

.

Lemma 13

If B = {k Ti A : k

x}, then |B| = O

x

(ln x)4

.


Our Research
http://find/


37/58



ln k20 ln ln k


If A =

{k

Ti : (

r

N)(r2

(ln x)10 and r2

|k)

}, then

|A

|=

O

x

(ln x)5

.

Lemma 13

If B = {k Ti A : k

x}, then |B| = O

x

(ln x)4

.

Lemma 14

If C = {k Ti (A B) : k

Prime Numbers and the Convergents of a Continued Fraction

Documents

Transcript of Prime Numbers and the Convergents of a Continued Fraction