Polynomial Identities

MATH 242 practice problems

Learning Goals: Develop and implement an algorithm to solve a mathematical problem, and interpret your results. Formulate precise mathematical conjectures supported by computational evidence. Explore generalizations of known mathematical ideas.

Your task: Complete Exercises 2.35 and 2.36 from our Experimental Mathematics text. These exercises are reprinted below.

Due: Monday, September 29, at 5pm. Upload your solution as a Mathematica notebook to the Polynomial Identities assignment on Moodle.

Grading: If you implement the requested functions and use them to compute the requested quantities, you will earn a Satisfactory grade.

Need help? Review the notes from class (which are posed on the course web site), talk with your classmates, email Prof. Wright, or visit the help session.

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Problem 1: Exercise 2.35, part 1

Write a function findFibCoeffs that implements Algorithm 2.17 on page 80 in the text. Verify that it works properly for the 3n and 5n cases. Specifically, confirm that:

findFibCoeffs(3) returns {5,0,3}

findFibCoeffs(5) returns {25,0,25, 0, 5}

Problem 2: Exercise 2.35, part 2

Using your function findFibCoeffs from the previous problem, find coefficients for the Fibonacci Q n identities for Q∈{7,9,11,...,25}. Report on any patterns that you observe.

Problem 3: Exercise 2.36

Write a function verifyFibIdentity that implements Algorithm 2.18 on page 81 in the text. Check whether your coefficients from the previous exercise produce identities that actually hold for even integers n up to 100 or more.