A Revıew of Mathematical Conjectures: Exploring Engaging Topics for University Mathematics Students

Abstract

Throughout history, humanity has been driven by an innate curiosity to explore beyond established boundaries, particularly evident in scientific and mathematical pursuits. The realm of mathematics has seen numerous conjectures spanning ancient times to the present day, encompassing various mathematical domains. These conjectures, some evolving into theorems upon proof, others being refuted and replaced, and a few remaining yet unresolved, form a significant facet of intellectual exploration. They captivate not only professional mathematicians but also enthusiasts, contributing to the evolution of mathematics. Mathematical conjectures are statements that have not yet been proven to be true or false. Typically created from observed patterns, these conjectures often originate from seemingly simple propositions. Presently, advancements in computer programming have substantially contributed to and aided in proving wrong by finding some counterexamples or confirming the conjectures for very large numbers. Python, in particular, facilitates the verification of conjectures for larger numbers, the identification of patterns and formulas, confirming conjectures or helping in finding counterexamples leading to rejection, as well as refining existing ones or generating new ones. The article aims to present several famous math conjectures, predominantly in number theory, and emphasize the importance and use of working with students for a more interesting class. Notable conjectures include Euclid's perfect number conjecture, Fermat's number conjecture, Collatz's conjecture, Landau’s conjecture, Mersenne's prime conjecture, and more.