Unsolved Conjectures in Number Theory: A Comprehensive Guide
Number theory is a vast and fascinating field in mathematics that remains rich with unsolved conjectures and problems. Here, we provide a guide to where you can find such problems, along with an intriguing conjecture by Gurthbruins.
Where to Find Unsolved Conjectures in Number Theory
The exploration of unsolved problems in number theory is not limited to one particular resource, but rather a combination of various reputable sources. From books and online resources to academic journals and mathematics communities, these platforms provide a comprehensive overview of the remaining puzzles in number theory.
Books
Reading books dedicated to number theory can help uncover unexplored territory. Notable works like 'Unsolved Problems in Number Theory' by Richard Guy offer a collection of intriguing problems and conjectures. These books often serve as a starting point for those delving into the mysteries that remain unsolved.
Online Resources
Online platforms offer interactive and dynamic ways to explore these conjectures. Wikipedia is a key resource, as it frequently updates with new information on current unsolved problems in mathematics, including number theory. Additionally, the American Mathematical Society (AMS) website features articles and papers discussing open problems in mathematics.
Research Papers and Journals
Academic journals play a crucial role in the dissemination of knowledge about unsolved problems in number theory. Look for papers in journals such as the Journal of Number Theory. These publications often publish surveys or discussions of open problems in the field, providing valuable insights and methods for addressing these conjectures.
Math Conferences and Workshops
Attending conferences and workshops is another way to stay updated on the latest research and open problems. These events often feature sessions dedicated to open problems in number theory. Proceedings from these conferences may also be published, making them valuable resources for anyone interested in these conjectures.
Online Mathematics Communities
Online communities such as MathOverflow or Stack Exchange are useful for discussions about unsolved problems and conjectures in number theory. These platforms allow mathematicians and enthusiasts to collaborate, share ideas, and discuss potential solutions to these conjectures.
A Specific Conjecture by Gurthbruins
Among the many unsolved problems in number theory, Gurthbruins' conjecture stands out for its unique and intriguing nature. The conjecture revolves around a Diophantine equation:
x2 - 2y2 - z xyz
Gurthbruins proposed that the following formulae for z and z lead to solutions:
z 22^n - 12 2n - 12
z 22^n 12 2n - 12
Bobbym, a member of the Math Is Fun Forum, tested this conjecture for n up to 50 and shared his findings in a series of posts. His last post stated:
'The problem is still computationally very difficult. The high n are achieved through a trick. I will post when I get to n 50. Also there is a little light concerning proving for any n. For n 50: x4802766836412989 digits, y9132345394403019 digits, z3802951800684685952690295930883'
For more information and the history of this conjecture, you can refer to discussions on Math Is Fun Forum and insights shared by other enthusiasts in the field.
Exploring these resources will not only deepen your understanding of number theory but also provide a roadmap for contributing to the solution of these outstanding problems.