Proving theorems and special cases: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on attempting to prove theorems by looking at special cases.

Famous propositions that are true for the first few cases (or many cases) before failing.

Part 1: The proposition “Is n^2 -n + 41 always a prime number?” is true for the first 40 cases but fails with n = 41.

Part 2: The Polya conjecture is true for over 900 million cases before failing.

Part 3: A conjecture about the distribution of prime numbers is true for the first 10^{318} cases before failing.

Mathematical induction

Part 4: Pedagogical thoughts on mathematical induction.

Part 5: More pedagogical thoughts on mathematical induction.

Famous unsolved problems in mathematics

Part 6: The Goldbach conjecture.

Part 7: The twin prime conjecture.

Part 8: The Collatz conjecture.

Part 9: The Riemann hypothesis.

Theorems in secondary mathematics that can be proven using special cases

Part 10: The sum of the angles in a convex polygon with n sides is 180n degrees.

Part 11: The Law of Cosines.

Part 12: Trig identities for \sin(\alpha \pm \beta) and \cos(\alpha \pm \beta).

Part 13: Uniqueness of logarithms.

Part 14: The Power Law for derivatives, or \displaystyle \frac{d}{dx} \left(x^n \right) = nx^{n-1}.

Part 15: The Mean Value Theorem.

Part 16: An old calculus problem of mine.

Part 17: Ending thoughts.


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