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Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof.
Significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually simple, whereas finding proofs (automated theorem proving) is generally hard. An informal proof in the mathematics literature, by contrast, requires weeks of peer review to be checked, and may still contain errors.
The proof of this is spread out over hundreds of journal articles which makes it hard to estimate its total length, which is probably around 10000 to 20000 pages. 2004 Robertson–Seymour theorem. The proof takes about 500 pages spread over about 20 papers. 2005 Kepler conjecture.
A propositional proof system is given as a proof-verification algorithm P(A,x) with two inputs.If P accepts the pair (A,x) we say that x is a P-proof of A.P is required to run in polynomial time, and moreover, it must hold that A has a P-proof if and only if A is a tautology.
Goldbach’s Conjecture. One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes ...
In any area of mathematics defined by its assumptions or axioms, a proof is an argument establishing a theorem of that area via accepted rules of inference starting from those axioms and from other previously established theorems. [7] The subject of logic, in particular proof theory, formalizes and studies the notion of formal proof. [8]
A proof that is based on new and original insights. A method of proof that can be easily generalized to solve a family of similar problems. In the search for an elegant proof, mathematicians may search for multiple independent ways to prove a result, as the first proof that is found can often be improved.