# Bachet`s Equation - Boise State University

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Bachet`s Equation - Boise State University

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.