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Bachet’s Equation
∆GSN
Clay McGowen
Scott Navert
Evan Smith
Boise State University
17 March 2015
Outline
Group Info
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications
References
Outline
Group Info
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications
References
Outline
Group Info
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications
References
Outline
Group Info
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications
References
Outline
Group Info
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications
References
Outline
Group Info
Data & Analysis
Elliptic Curves & Prime factorization
Financial Applications
References
Background
Bachet’s equation is a famous Diophantine equation that looks
like:
y 2 = x 3 + k for k ∈ Z
Background
Bachet’s equation is a famous Diophantine equation that looks
like:
y 2 = x 3 + k for k ∈ Z
Data 01
Shout out to Chuck and Project B for sharing their data
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Zp k |S| |S| ∪ idg
2 1 2
3
3 1 3
4
3 2 3
4
5 1 5
6
5 2 5
6
5 3 5
6
5 4 5
6
Table: Data
Data 01
Shout out to Chuck and Project B for sharing their data
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Zp k |S| |S| ∪ idg
2 1 2
3
3 1 3
4
3 2 3
4
5 1 5
6
5 2 5
6
5 3 5
6
5 4 5
6
Table: Data
Data 01
Shout out to Chuck and Project B for sharing their data
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Zp k |S| |S| ∪ idg
2 1 2
3
3 1 3
4
3 2 3
4
5 1 5
6
5 2 5
6
5 3 5
6
5 4 5
6
Table: Data
Data 01
Shout out to Chuck and Project B for sharing their data
We looked at integers < p where p is prime. We also looked
at an identity element for each group.
Zp k |S| |S| ∪ idg
2 1 2
3
3 1 3
4
3 2 3
4
5 1 5
6
5 2 5
6
5 3 5
6
5 4 5
6
Table: Data
Data 02
What does this look like?
In Z5 :
x
4
0
2
2
0
y
0
1
2
3
4
Table: Sk=1
x
1
3
0
0
3
y
0
1
2
3
4
Table: Sk=4
Data 02
What does this look like?
In Z5 :
x
4
0
2
2
0
y
0
1
2
3
4
Table: Sk=1
x
1
3
0
0
3
y
0
1
2
3
4
Table: Sk=4
Data 02
What does this look like?
In Z5 :
x
4
0
2
2
0
y
0
1
2
3
4
Table: Sk=1
x
1
3
0
0
3
y
0
1
2
3
4
Table: Sk=4
Data 02
What does this look like?
In Z5 :
x
4
0
2
2
0
y
0
1
2
3
4
Table: Sk=1
x
2
4
3
3
4
y
0
1
2
3
4
Table: Sk =2
x
3
2
1
1
2
y
0
1
2
3
4
Table: Sk =3
x
1
3
0
0
3
y
0
1
2
3
4
Table: Sk=4
Data 03
What if |S| =
6 p?
In Z7 :
Zp
7
7
7
7
7
7
k
1
2
3
4
5
6
|S|
11
8
12
2
6
3
|S| ∪ ID
12
9
13
3
7
4
Table: Data Z7
x
1
2
4
y
0
0
0
Table: Sk =6
Data 03
What if |S| =
6 p?
In Z7 :
Zp
7
7
7
7
7
7
k
1
2
3
4
5
6
|S|
11
8
12
2
6
3
|S| ∪ ID
12
9
13
3
7
4
Table: Data Z7
x
1
2
4
y
0
0
0
Table: Sk =6
Data 03
What if |S| =
6 p?
In Z7 :
Zp
7
7
7
7
7
7
k
1
2
3
4
5
6
|S|
11
8
12
2
6
3
|S| ∪ ID
12
9
13
3
7
4
Table: Data Z7
x
1
2
4
y
0
0
0
Table: Sk =6
Data 03
What if |S| =
6 p?
In Z7 :
Zp
7
7
7
7
7
7
k
1
2
3
4
5
6
|S|
11
8
12
2
6
3
|S| ∪ ID
12
9
13
3
7
4
Table: Data Z7
x
1
2
4
y
0
0
0
Table: Sk =6
Data 03
What if |S| =
6 p?
In Z7 :
Zp
7
7
7
7
7
7
k
1
2
3
4
5
6
|S|
11
8
12
2
6
3
|S| ∪ ID
12
9
13
3
7
4
Table: Data Z7
x
1
2
4
y
0
0
0
Table: Sk =6
Data 04
Results of this investigation:
Conjecture: 1
If |S| = p
Then p is a prime of the form 3n − 1 and 6n − 1 where n ∈ Z.
Conjecture: 2
If |S| =
6 p
Then p is an elliptic prime.
Data 04
Results of this investigation:
Conjecture: 1
If |S| = p
Then p is a prime of the form 3n − 1 and 6n − 1 where n ∈ Z.
Conjecture: 2
If |S| =
6 p
Then p is an elliptic prime.
Data 04
Results of this investigation:
Conjecture: 1
If |S| = p
Then p is a prime of the form 3n − 1 and 6n − 1 where n ∈ Z.
Conjecture: 2
If |S| =
6 p
Then p is an elliptic prime.
Data 04
Results of this investigation:
Conjecture: 1
If |S| = p
Then p is a prime of the form 3n − 1 and 6n − 1 where n ∈ Z.
Conjecture: 2
If |S| =
6 p
Then p is an elliptic prime.
Elliptic Curve Factoring
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Elliptic Curve Factoring
We will start with a random, composite, odd integer n that
we wish to factor. We will then perform the following steps.
[1]
1. Choose several random elliptic curves
Ei : y 2 = x 3 + Ai x + Bi (usually around 10 to 20) and points
Pi mod n. [1]
2. Choose an integer k (for example 108 ) and compute
(k!)Pi on Ei for each i. [1]
3. If step 2 fails because some slope does not exist mod n,
then we have found a factor of n. [1]
4. If step 2 succeeds, increase k or choose new random
curves Ei and points Pi and start over. [1]
Primality Testing
Elliptic curves can be used to find higher order primes.
Primality Testing
Elliptic curves can be used to find higher order primes.
Further Applications
With our development of Elliptic Primes, we need to investigate
their applications in further research. Specifically, we will look at:
Determining the Primality of High Order Numbers
Applications in Financial Retracement
Game Theory in Addition Games over Zp
Elliptic Curve Cryptogaphy
Further Applications
With our development of Elliptic Primes, we need to investigate
their applications in further research. Specifically, we will look at:
Determining the Primality of High Order Numbers
Applications in Financial Retracement
Game Theory in Addition Games over Zp
Elliptic Curve Cryptogaphy
Financial Applications
Mathematics and Finance are incredibly intertwined.
”Rocket Scientist” Mathematicians are fueling incredible
amounts of trades with data interpreted through the scope
of mathematics.
Currently, there exists no formal literature on the application
of Elliptic Primes in the financial hemisphere. This may be
because the concept is relatively new, or that financial
institutions are reluctant to unproven forms of mathematics
Financial Applications
Mathematics and Finance are incredibly intertwined.
”Rocket Scientist” Mathematicians are fueling incredible
amounts of trades with data interpreted through the scope
of mathematics.
Currently, there exists no formal literature on the application
of Elliptic Primes in the financial hemisphere. This may be
because the concept is relatively new, or that financial
institutions are reluctant to unproven forms of mathematics
Financial Applications
A Quick Primer and Definitions:
Stock - A share of a company that is publicly traded on an
Exchange
Option - A contract between two parties usually on the
purchase price of a stock or commodity (ie, someone has
the option to purchase something at a previously agreed
upon value)
Price Level - The current market price for a stock or
commodity
Entry/Exit Price The price level for which an individual would
enter/exit the market
Efficient Market Hypothesis, EMH - The idea that markets
are self correcting and adjust themselves perfectly to the
availability of information
Financial Applications
A Quick Primer and Definitions:
Stock - A share of a company that is publicly traded on an
Exchange
Option - A contract between two parties usually on the
purchase price of a stock or commodity (ie, someone has
the option to purchase something at a previously agreed
upon value)
Price Level - The current market price for a stock or
commodity
Entry/Exit Price The price level for which an individual would
enter/exit the market
Efficient Market Hypothesis, EMH - The idea that markets
are self correcting and adjust themselves perfectly to the
availability of information
Financial Applications
A technique commonly used by mathematicians on Wall
Street to analyze entry and exit prices is the Fibonacci
Retracement method. This method takes an application of
the Fibonacci numbers to track stock prices over time.
Recall that general ratio of two Fibonacci numbers is 1.618.
This ratio will be important to our application. The general
purpose of our retracement is to graph ratios of Fibonacci
numbers to the stock prices over a certain time, and isolate
the supports and resistances to price fluctuations.
Financial Applications
A technique commonly used by mathematicians on Wall
Street to analyze entry and exit prices is the Fibonacci
Retracement method. This method takes an application of
the Fibonacci numbers to track stock prices over time.
Recall that general ratio of two Fibonacci numbers is 1.618.
This ratio will be important to our application. The general
purpose of our retracement is to graph ratios of Fibonacci
numbers to the stock prices over a certain time, and isolate
the supports and resistances to price fluctuations.
Financial Applications
A technique commonly used by mathematicians on Wall
Street to analyze entry and exit prices is the Fibonacci
Retracement method. This method takes an application of
the Fibonacci numbers to track stock prices over time.
Recall that general ratio of two Fibonacci numbers is 1.618.
This ratio will be important to our application. The general
purpose of our retracement is to graph ratios of Fibonacci
numbers to the stock prices over a certain time, and isolate
the supports and resistances to price fluctuations.
Financial Applications
Financial Applications
The following ratios are calculated for our application:
F100 = 31.24
F61.8 = 29.85
F50 = 29.43
F38.2 = 29.00
F0 = 27.61
Financial Applications
And now, we superimpose the ratios on the graph:
Financial Applicaiton
I noticed how these primes are similar to the ratios between the
Fibonacci Numbers. So, we calculate the elliptic primes ratios
like so:
En /En−1 = 13/7 − 1 ≈ 0.85
En+1 /En = 19/7 − 1 ≈ 0.46
0.85/0.46 = 1.27
Our ratios will be 0, 100, 27, 46, and 85
Financial Applications
Items to further invesitgate:
Risk Analysis over time
FOREX and Option markets; derivative commodities
Price Volatility
Random Walk design
References
Lawrence C. Washington. Elliptic Curves: Number Theory
and Cryptography. Chapman & Hall/CRC, Boca Raton,
Florida, 2008.