Unsolved Problems
Tim Roberts maintains the website http://www.unsolvedproblems.org/ in which he presents unsolved problems from mathematics, mostly conjectures that yet lack a formal proof. While working on these problems is fun, it may be useful to know that some problems might be unsolvable, so they are not worth the effort trying to solve them. By means of formal logic we can show that some problems are probably unsolvable. If we can reformulate a statement to be of the type "there does not exist any n for which ...", it is probably unprovable. Examples: Goldbach Conjecture The Goldbach Conjecture states that every even number greater than 2 is the sum of two primes. A number is prime if it is divisible only by itself and 1. So, for example, 36 = 17+19. This is equivalent to: For every integer n greater than 1, there is an integer k so that both n - k and n + k are prime numbers. Employing the existential quantifier only, we can reformulate this to: There does not exist any integer n grea