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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. However, outside the field of automated proof assistants, this is rarely done in practice.
The proof [3] is based on a fact that a semigroup S is finitely generated if and only if its semigroup algebra [] is a finitely generated algebra over . To prove Gordan's lemma, by induction (cf. the proof above), it is enough to prove the following statement: for any unital subsemigroup S of Z d {\displaystyle \mathbb {Z} ^{d}} ,
An archetypal double counting proof is for the well known formula for the number () of k-combinations (i.e., subsets of size k) of an n-element set: = (+) ().Here a direct bijective proof is not possible: because the right-hand side of the identity is a fraction, there is no set obviously counted by it (it even takes some thought to see that the denominator always evenly divides the numerator).
It is an important proof technique in set theory, topology and other fields. Proofs by transfinite induction typically distinguish three cases: when n is a minimal element, i.e. there is no element smaller than n; when n has a direct predecessor, i.e. the set of elements which are smaller than n has a largest element;
The classical proof is straightforward but relies on Ado's theorem, whose proof is algebraic and highly non-trivial. [2] Ado's theorem states that any finite-dimensional Lie algebra can be represented by matrices. As a consequence, integrating such algebra of matrices via the matrix exponential yields a Lie group integrating the original Lie ...
Every textbook on elementary number theory (and quite a few on algebraic number theory) has a proof of quadratic reciprocity. Two are especially noteworthy: Lemmermeyer (2000) has many proofs (some in exercises) of both quadratic and higher-power reciprocity laws and a discussion of their history. Its immense bibliography includes literature ...
This is a list of unusually long mathematical proofs.Such proofs often use computational proof methods and may be considered non-surveyable.. As of 2011, the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10000 pages.
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