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/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
3/58
Introduction
Continued fractions have applications in
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction
3 / 30
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
4/58
Introduction
Continued fractions have applications in
1 Mathematical Cryptography [1].
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 3 / 30
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
5/58
Introduction
Continued fractions have applications in
1 Mathematical Cryptography [1].
2 Atomic Physics [2].
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 3 / 30
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
6/58
7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
7/58
Introduction
Continued fractions have applications in
1 Mathematical Cryptography [1].
2 Atomic Physics [2].
3 Cosmology [3].
4 Ecology [4].
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 3 / 30
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
8/58
Introduction
Continued fractions have applications in
1 Mathematical Cryptography [1].
2 Atomic Physics [2].
3 Cosmology [3].
4 Ecology [4].
The security of many cryptosystems rely on a computationally difficultmathematical problem (see eg. [1], [5], [6]).
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 3 / 30
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 4 / 30
http://find/http://goback/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
10/58
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.
1 Wieners Method [7].
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 4 / 30
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
11/58
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.
1 Wieners Method [7].
2 CFRAC [8].
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 4 / 30
http://find/http://goback/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 5 / 30
http://find/http://goback/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 6 / 30
http://find/http://goback/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 6 / 30
http://find/http://goback/7/30/2019 Prime Numbers and the Convergents of a 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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 7 / 30
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 7 / 30
C
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
17/58
Convergents
We also have some very nice properties that involve the convergents of acontinued fraction.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 8 / 30
C
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 8 / 30
C
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 9 / 30
C t
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
20/58
Convergents
Theorem 4 (Dirichlet, [12])Let x be irrational, then there are infinitely many rationals numbers A
B
such that
xA
B 1
B2 .
Manufacturers will sometimes create cryptosystems with backdoors as anyeasy way to gain access in the event that a private key is lost.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 9 / 30
C t
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
21/58
Convergents
Theorem 4 (Dirichlet, [12])Let x be irrational, then there are infinitely many rationals numbers A
B
such that
xA
B 1
B2 .
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].
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 9 / 30
Convergents
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 10 / 30
Big O Notation
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 11 / 30
Big O Notation
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
24/58
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.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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 11 / 30
Big O Notation
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
25/58
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.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.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))).
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 11 / 30
Big O Notation
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
26/58
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.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.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)).
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 11 / 30
Motivation
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
27/58
Motivation
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 12 / 30
Motivation
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 12 / 30
Motivation
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 12 / 30
Our Research
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
30/58
Our Research
In our research we focus on the numerator An instead of Bn.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 13 / 30
Our Research
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 13 / 30
Our Research
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
32/58
Denote G(k) as the greatest prime factor of the number k.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 14 / 30
Our Research
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
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 .
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 14 / 30
Our Research
http://find/http://goback/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
34/58
We give a much more detailed description of Theorem 11 by splitting it up
into three Lemmas.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 15 / 30
Our Research
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
35/58
We give a much more detailed description of Theorem 11 by splitting it up
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
.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 15 / 30
Our Research
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
36/58
We give a much more detailed description of Theorem 11 by splitting it up
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
.
Lemma 13
If B = {k Ti A : k
x}, then |B| = O
x
(ln x)4
.
C. Humphreys (BSU) Prime Numbers and the Convergents of a Continued Fraction 15 / 30
Our Research
http://find/7/30/2019 Prime Numbers and the Convergents of a Continued Fraction
37/58
We give a much more detailed description of Theorem 11 by splitting it up
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
.
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