1. Diophantine Equations Group 2: Michelle Levitsky, Kai Yin Lee, Stephanie Loo, Rouge Yang

2. Aim: What are Diophantine Equations and how do we solve them?

3. an equation involving two or more variables in which the coefficients of the variables and solutions to the problem are integers

4. Examples of Diophantine Equations (Linear Diophantine Equation) (If n = 2, there are an infinite amount of solutions for x, y, and z, the Pythagorean Triples. For larger values of n, Fermat’s Last Theorem states that there are no positive integer solutions for x, y, and z satisfying this equation) (Pell’s equation) (The Erdös-Straus conjecture, states that for every positive integer n ≥ 2, there is a solution for x, y, and z as positive integers.)

5. Infinite Diophantine Equations This equation always has a solution for a positive n.

6. Exponential Diophantine Equations if a Diophantine equation has another variable or variables in the form of exponents, it is an exponential Diophantine equation

7. Solving a Diophantine Equation: 1) Identify the variable for which you want to solve. Regardless of the variable you choose, the solution method is similar due to the symmetry involved in the equation. 2) Subtract the products of the variables not chosen and their coefficients from both sides of the equation. For example, if you want to solve for z, you need to subtract the products ax and by from both sides of the equation. In this example, you would be left with the new equation cz = d - ax - by. 3) Divide both sides of the equation by the coefficient for the variable that you are solving for. In the example, the coefficient multiplying z is c. Therefore, you would divide both sides of the equation by c, resulting in z = d / c – ( a / c ) x – ( b / c ) y. This is the solution to the equation for the variable z.

8. 1. Practice

9. 2.You want to form 83 cents in postage using only 6- and 15- cent stamps. How many of each type should be used?

10. 3.Albert wants to purchase $510 in traveler’s checks using only $20 and $50 checks. How many of each type of check should he purchase?

11. 4.A grocer orders apples and pears at a total cost of $8.39. If she pays 25 cents per apple and 18 cents per pear, how many of each type did she order?

12. 5.Using only 14- and 21-cent stamps, how would you form postage of $3.00?

13. 6.Chris bought a dozen pieces of fruit, apples and bananas, for $1.32. If an apple costs 3 cents more than a banana and more apples are purchased than bananas, how many pieces of each type of fruit did Swann buy?