On the strategies used to attack unsolved mathematical problems - a case study of the Collatz Conjecture

Amir Zoet, Johan Mogens Følsgaard, Rasmus Kristoffer Pedersen & Kasper Emil Dueholm Freiman

Studenteropgave: Semesterprojekt

Abstrakt

Mathematical problems can be easy to state, yet very difficult to solve. A famous example of this is known as the Collatz Conjecture. In this report a collection of articles related to the Collatz Conjecture has been surveyed in order to gain insight into mathematical research in general. %Maybe remove from "to gain ins..." to the end? From a thorough study of the collection of articles we identified the most significant results and theorems. These were then analyzed to figure out which strategy was used to work on the conjecture. %"figure out"? In our study we focused on five main strategies; reduction, reformulation, restriction and generalization of the problem as well as migration to another mathematical area. Summarizing the results we found that there has indeed been found many important characteristics about the function with which the Collatz Conjecture is concerned. Yet none of these provide results that make settling the Collatz Conjecture easy. However, the vast amount of literature on the subject provides an important basis for future research and anyone who hopes to settle the conjecture should do a preliminary study of previous results. %before starting på en pæn måde The strategies used to categorize the results were found to be adequate, although many approaches to the problem were found to use multiple strategies to some extent. %extend eller extent? We believe that a similar categorization could be used for other problems. Researching the Collatz Conjecture required us to study a wide range of mathematical areas as well as learning how mathematical research is done on such problems. From this we find that the Collatz Conjecture worked sufficiently as a case for studying mathematical research in general.

UddannelserMatematik, (Bachelor/kandidatuddannelse) Bachelor el. kandidat
SprogEngelsk
Udgivelsesdato22 jan. 2016
VejledereMogens Allan Niss

Emneord

  • The Collatz Conjecture
  • Kakutani's problem
  • The Ulam problem
  • Hailstone sequence
  • Hailstone numbers
  • Thwaite's conjecture
  • The 3x + 1 conjecture
  • The Syracuse problem
  • Hasse's algorithm
  • The 3n + 1 conjecture