History of mathematics · Maths EE idea · Accessible
Egyptian fractions
A research question to start from
Why can every fraction be written as a sum of distinct unit fractions, and how efficient is the greedy method?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
An ancient practice with a modern proof and algorithm analysis.
Mathematics you would need
- Unit fractions
- The greedy algorithm
- Proof of termination
- Bounds on denominators
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Describe the historical use briefly.
- Prove the greedy algorithm terminates.
- Analyse how large denominators can get and compare with other methods.
Scope and difficulty
Accessible. Accessible.
Pitfalls
- Too much history.
- Termination not proved.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Egyptian fractions greedy algorithm proof Fibonacci-Sylvester. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).