A mathematical theorem is a statement that has been proven to be true based on previously established statements, such as axioms and other theorems. Theorems are essential in mathematics because they provide a foundation for further exploration and understanding. For example, the Pythagorean Theorem helps us find the length of a side in a right triangle, showing the relationship between the sides. To prove a theorem, mathematicians use logical reasoning and evidence. This process often involves constructing a proof, which is a series of statements that demonstrate the truth of the theorem. Famous theorems, like Fermat's Last Theorem, have challenged mathematicians for centuries, showcasing the depth and complexity of mathematical inquiry.