What is a theorem?
Q: What is a theorem?
A: A theorem is an idea that has been proven to be true in mathematics using logic and other theorems that have already been proved.
Q: What is a lemma?
A: A lemma is a minor theorem that one must prove to prove a major theorem.
Q: How are theorems made up?
A: Theorems are made of two parts - hypotheses and conclusions - and use deduction rather than empirical theories.
Q: Are all theorems difficult to prove?
A: No, some theorems are trivial since they directly follow from propositions, while others require long and difficult proofs which involve other areas of mathematics or show connections between different areas.
Q: Can a theorem be simple yet deep?
A: Yes, an example of this would be Fermat's Last Theorem which is simple to state but its proof is long and difficult.
Q: Are there any theorems for which a proof is known but cannot easily be written down?
A: Yes, examples include the four color theorem and Kepler conjecture which can only be verified by running them through computer programs.
Q: Can mathematical theorems sometimes be reduced to simpler computations?
A: Yes, mathematical theorems can sometimes be reduced to simpler computations such as polynomial identities, trigonometric identities or hypergeometric identities.
