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Euclid's proof of the fundamental theorem of arithmetic is a simple proof which uses a minimal counterexample. [5] [6] Courant and Robbins used the term minimal criminal for a minimal counter-example in the context of the four color theorem. [7]
The first complete proof was given by Hilbert in 1896. In 1879, Alfred Kempe published a purported proof of the four color theorem, whose validity as a proof was accepted for eleven years before it was refuted by Percy Heawood. Peter Guthrie Tait gave another incorrect proof in 1880 which was shown to be incorrect by Julius Petersen in 1891.
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy . [ 1 ] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and ...
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem ), which proves the existence of a particular kind of object ...
The conjecture was disproved in 1966, with a counterexample involving a count of only four different 5th powers summing to another fifth power: 27 5 + 84 5 + 110 5 + 133 5 = 144 5. Proof by counterexample is a form of constructive proof, in that an object disproving the claim is exhibited.
The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. To gain acceptance, a proof has to meet communal standards of rigor; an argument considered vague or incomplete may be rejected. The concept of proof is formalized in the field of mathematical logic. [12]
In logic and mathematics, proof by example (sometimes known as inappropriate generalization) is a logical fallacy whereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledged proof. [1] [2] The structure, argument form and formal form of a proof by example generally proceeds as follows ...
If is an odd prime and <, then (,) =. Proof: There are exactly two factors of in the numerator of the expression () = ()! / (!), coming from the two terms and in ()!, and also two factors of in the denominator from one copy of the term in each of the two factors of !.