is the proof of completeness for a complexity class, which provides evidence for the difficulty of problems by showing them to be just as hard to solve as any other problem in the class. In particular, a complete problem is intractable if one of the problems in its class is. Types of proof By contradiction One widely used type of impossibility proof is proof by contradiction. In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive. Since the contradiction stems from the original assumption, this means that the assumed premise must be impossible. By descent One type of proof by contradiction is proof by descent, which proceeds first by assuming that something is possible, such as a positive integer solution to a class of equations, and that therefore there must be a smallest solution (by the Well-ordering principle). From the alleged smallest solution, it is then shown that a smaller solution can be found, contradicting the premise that the former solution was the smallest one possible—thereby showing that the original premise that a solution exists must be false. Types of disproof There are two alternative methods of disproving a conjecture that something is impossible: by counterexample (constructive proof) and by logical contradiction (non-constructive proof). The obvious way to disprove an impossibility conjecture is by providing a single counterexample. For example, Euler proposed that at least n different nth powers were necessary to sum to yet another nth power. The conjecture was disproved in 1966, with a counterexample involving a count of only four different 5th powers summing to another fifth power: 275 + 845 + 1105 + 1335 = 1445. Proof by counterexample is a constructive proof, in that an object disproving the claim is exhibited. In contrast, a non-constructive proof of an impossibility claim would proceed by showing it is logically contradictory for all possible counterexamples to be invalid: at least one of the items on a list of possible counterexamples must actually be a valid counterexample to the impossibility conjecture. For example, a conjecture that it is impossible for an irrational power raised to an irrational power to be rational was disproved, by showing that one