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Fundamental theorem of arithmetic. Gauss–Markov theorem (brief pointer to proof) Gödel's incompleteness theorem. Gödel's first incompleteness theorem. Gödel's second incompleteness theorem. Goodstein's theorem. Green's theorem (to do) Green's theorem when D is a simple region. Heine–Borel theorem.
Mathematical fallacy. In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known ...
2 + 2 = 5 or two plus two equals five is a mathematical falsehood which is used as an ... when those who dared to say that 2 + 2 = 4 rather than 2 + 2 = 5 were ...
Pythagorean identities. Identity 1: The following two results follow from this and the ratio identities. To obtain the first, divide both sides of by ; for the second, divide by . Similarly. Identity 2: The following accounts for all three reciprocal functions. Proof 2: Refer to the triangle diagram above.
Then P(n) is true for all natural numbers n. For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P(n) represent " 2n − 1 is odd": (i) For n = 1, 2n − 1 = 2 (1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true.
Dirichlet's proof for n = 5 is divided into the two cases (cases I and II) defined by Sophie Germain. In case I, the exponent 5 does not divide the product xyz. In case II, 5 does divide xyz. Case I for n = 5 can be proven immediately by Sophie Germain's theorem(1823) if the auxiliary prime θ = 11.
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A triangle whose side lengths are a Pythagorean triple is a right triangle and ...
Glossary of mathematical symbols. A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various ...