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Gödel's second incompleteness theorem states no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. Stated more colloquially, any ...

A Hessenberg decomposition is a matrix decomposition of a matrix A into a unitary matrix P and a Hessenberg matrix H such that PHP^(H)=A, where P^(H) denotes the conjugate ...

A function defined for all positive integers, sometimes also called an arithmetic function (Nagell 1951, p. 26) or number theoretic function (Wilf 1994, p. 58).

A quantity which remains unchanged under certain classes of transformations. Invariants are extremely useful for classifying mathematical objects because they usually reflect ...

The German mathematician Kronecker proved that all the Galois extensions of the rationals Q with Abelian Galois groups are subfields of cyclotomic fields Q(mu_n), where mu_n ...

With a large enough sample, any outrageous thing is likely to happen (Diaconis and Mosteller 1989). Littlewood (1986) considered an event which occurs one in a million times ...

Linnik's constant L is the constant appearing in Linnik's theorem. Heath-Brown (1992) has shown that L<=5.5, and Schinzel, Sierpiński, and Kanold (Ribenboim 1989) have ...

Let A={a_1,a_2,...} be a free Abelian semigroup, where a_1 is the identity element, and let mu(n) be the Möbius function. Define mu(a_n) on the elements of the semigroup ...

A type of mathematical result which is considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Finite ...

The Nørlund polynomial (note that the spelling Nörlund also appears in various publications) is a name given by Carlitz (1960) and Adelberg (1997) to the polynomial ...