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Raise the Roof for Fibonacci’s Perfect Power

Raise the Roof for Fibonacci’s Perfect Power Courtney Killian, Tricia Stoner, Marielle Silvio, and Kasey Zemba California University of Pennsylvania – RMA. PROOFS. INTRODUCTION. Useful Identities. B-1111 Proposed by Mircea Merca , University of Craiova, Romania

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Raise the Roof for Fibonacci’s Perfect Power

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  1. Raise the Roof for Fibonacci’s Perfect Power Courtney Killian, Tricia Stoner, Marielle Silvio, and Kasey Zemba California University of Pennsylvania – RMA PROOFS INTRODUCTION Useful Identities B-1111 Proposed by MirceaMerca, University of Craiova, Romania Proved by: MarielleSilvio and Kasey Zemba The Proof (B-1111 Continued) • Fibonacci numbers play an important role in many areas of science, including mathematics, biology, combinatorics, formal languages, and more. • The purpose of this project was to attempt to prove proposed mathematical equations that were proposed in the August 2012 edition of the academic journal, The Fibonacci Quarterly. Useful Lemma Useful Definitions and Notation BACKGROUND • Fibonacci sequence was discovered by Leonardo Pisano Bigollo, a famous Italian mathematician. • The Fibonacci sequence states that the next number in the sequence is the sum of the previous two. • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. • Represented by the expression F1=1, F2=1, and Fn=Fn-1+Fn-2 for every n ≧3. • Fibonacci numbers can be found in nature • Number of petals on a flower • Number of pairs of rabbits that can be produced from a single pair of rabbits in a year The Proof Useful Theorems B-1112 Proposed by Jose Luis Diaz-Barrero, Barcelona Tech, Barcelona, Spain Proved by: Courtney Killian and Tricia Stoner • Golden Ratio • Ratio of sum of 2 quantities to the larger quantity is equal to the ratio of the larger quantity to smaller quantity CONCLUSIONS • Solutions submitted in January 2013 to The Fibonacci Quarterly • Currently under review by anonymous professional mathematicians • One original proof will be highlighted in the journal • By November 2013 • Plans exist to submit more original proofs to proposed and open problems in the future. Useful Theorems THE FIBONACCI ASSOCIATION • “The Fibonacci Association, incorporated in 1963, focuses on Fibonacci numbers and related mathematics, emphasizing new results, research proposals, challenging problems, and new proofs of old ideas.” • Releases a quarterly academic journal, The Fibonacci Quarterly, which contains • University level articles • Proposed problems and solutions The Proof REFERENCES • R. L. Grahm, D. E. Knuth, O. Patashinik, Concrete Mathematics, Addison-Wesley, New York, 1990. • T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley, New York, 2001. • The On-Line Encyclopedia of Integer Sequences, January 3 2013. http://oeis.org/A001175.

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