For example, there are 20, 138, 200 Carmichael numbers between 1 and 10 < sup > 21 </ sup > ( approximately one in 50 billion numbers ).< ref name =" Pinch2007 "> Richard Pinch, " The Carmichael numbers up to 10 < sup > 21 </ sup >", May 2007 .</ ref > This makes tests based on Fermat's Little Theorem slightly risky compared to others such as the Solovay-Strassen primality test.

Advancements in technology have led to significant improvement in efficiencies, as well as a greater variety of methods to test the Bell Theorem.

: Theorem ( A. Korselt 1899 ): A positive composite integer is a Carmichael number if and only if is square-free, and for all prime divisors of, it is true that ( where means that divides ).

An interesting characteristic of the Blum Blum Shub generator is the possibility to calculate any x < sub > i </ sub > value directly ( via Euler's Theorem ):

" A Completeness Theorem in Modal Logic ", Journal of Symbolic Logic 24 ( 1 ): 1 – 14.

Corollary ( Pointwise Ergodic Theorem ): In particular, if T is also ergodic, then is the trivial σ-algebra, and thus with probability 1:

* Theorem ( by Mackey ): Given a dual pair, the bounded sets under any dual topology are identical.

Assume p and q − 1 are relatively prime, a similar application of Fermat's Little Theorem says that ( q − 1 )< sup >( p − 1 )</ sup > ≡ 1 ( mod p ).

: Theorem: Assume T is a bounded linear operator from L < sup > p </ sup > to L < sup > p </ sup > and at the same time from L < sup > q </ sup > to L < sup > q </ sup >.

Theorem: Assume T is a bounded linear operator from to and at the same time from to.

: Wernicke's Theorem: Assume G is planar, nonempty, has no faces bounded by two edges, and has minimum degree 5.

Theorem: K is not a computable function.

: Turing's thesis: " Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i. e. by one of his machines, is equivalent to Church's thesis by Theorem XXX.

: Theorem on projections: Let the function f: X → B be such that a ~ b → f ( a )

The Second Main Theorem, more difficult than the first one tells that there are relatively few values which the function assumes less often than average.

" The Implicit Function Theorem states that if is defined on an open disk containing, where,, and and are continuous on the disk, then the equation defines as a function of near the point and the derivative of this function is given by ..."

Tautology -- Temporal logic -- Term -- Term logic -- Ternary logic -- Theorem -- Tolerance -- Trilemma -- Truth -- Truth condition -- Truth function -- Truth value -- Type theory

This theorem is called the Second Fundamental Theorem of Nevanlinna Theory, and it allows to give an upper bound for the characteristic function in terms of N ( r, a ).

The Second Fundamental Theorem implies that the set of deficient values of a function meromorphic in the plane is at most countable and the following relation holds:

Theorem For ƒ in L < sup > 1 </ sup >( R < sup > d </ sup >), the above series converges pointwise almost everywhere, and thus defines a periodic function Pƒ on Λ. Pƒ lies in L < sup > 1 </ sup >( Λ ) with || Pƒ ||< sub > 1 </ sub > ≤ || ƒ ||< sub > 1 </ sub >.

Theorem: A group decision function with an odd number of voters meets conditions 1, 2, 3, and 4 if and only if it is the simple majority method.

* Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud

* Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud

However, unlike the value of most other assets, the value of land is largely a function of government spending on services and infrastructure ( a relationship demonstrated by economists in the Henry George Theorem ).

Schwarz's Theorem then provides us with a necessary criterion for the existence of a potential function.

The universality of the Riemann zeta function was first proven by Sergei Mikhailovitch Voronin in 1975 and is sometimes known as Voronin's Universality Theorem.

Theorem For any normalized continuous positive definite function f on G ( normalization here means f is 1 at the unit of G ), there exists a unique probability measure on such that

so the First Fundamental Theorem says that the sum N ( r, a, f ) + m ( r, a, f ), tends to infinity at the rate which is independent of a.

By the First Fundamental Theorem, 0 ≤ δ ( a, f ) ≤ 1, if T ( r, f ) tends to infinity ( which is always the case for non-constant functions meromorphic in the plane ).

Theorem Every self-adjoint f in A * can be written as f

* Although Sard's Theorem does not hold in general, every continuous map f: X → R < sup > n </ sup > from a Hilbert manifold can be arbitrary closely approximated by a smooth map g: X → R < sup > n </ sup > which has no critical points

Unicity: Suppose satisfies, then by Theorem 1. 8,.

General Impossibility Theorem: It is impossible to formulate a social preference ordering that satisfies all of the following conditions:

Theorem: Let V be a finite-dimensional vector space over a field F, and A a square matrix over F. Then V ( viewed as an F-module with the action of x given by A and extending by linearity ) satisfies the F-module isomorphism

In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

* Proof of Wedderburn's Theorem at Planet Math

Image: Thales ' Theorem Simple. svg | Thales ' theorem: if AC is a diameter, then the angle at B is a right angle.

* GCD and the Fundamental Theorem of Arithmetic at cut-the-knot

He was appointed a lecturer in mathematics at Cambridge in 1927, where his 1935 lectures on the Foundations of Mathematics and Gödel's Theorem inspired Alan Turing to embark on his pioneering work on the Entscheidungsproblem ( decision problem ) using a hypothetical computing machine.

Case 2 includes all p that divide at least one of x, y, or z. Germain proposed the following, commonly called “ Sophie Germain's Theorem ”:

On the other hand, the scholars invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of prime numbers into a physical question.

Notable films from this period include: La Dolce Vita, 8½ ; La Notte ; L ' Eclisse, The Red Desert ; Blowup ; Satyricon ; Accattone ; The Gospel According to St. Matthew ; Theorem ; Winter Light ; The Silence ; Persona ; Shame ; A Passion ; Au Hasard Balthazar ; Mouchette ; Last Year at Marienbad ; Chronique d ' un été ; Titicut Follies ; High School ; Salesman ; La jetée ; Warrendale ; Knife in the Water ; Repulsion ; The Saragossa Manuscript ; El Topo ; A Hard Day's Night ; and the cinema verite Dont Look Back.

*" The Surprise Examination Paradox and the Second Incompleteness Theorem " by Shira Kritchman and Ran Raz, at ams. org

His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NP-Completeness.

* Fermat's Little Theorem at cut-the-knot

* Euler Function and Theorem at cut-the-knot

* Gauss – Bonnet Theorem at Wolfram Mathworld

At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents using what we now recognize as the fact that the ring is a Dedekind domain.

* Bezout's Theorem at MathPages

* Derivations and applications of Ceva's Theorem at cut-the-knot

* Trigonometric Form of Ceva's Theorem at cut-the-knot

* Abel's Impossibility Theorem at Everything2

* Proof of Dehn's Theorem at Everything2

However, the link between the Riemann hypothesis and the Prime Number Theorem had been known before in Continental Europe, and Littlewood also wrote later in his book A mathematician ’ s miscellany that his actually only rediscovered result did not shed a bright light on the isolated state of British mathematics at the time.

* Feuerbach's Theorem: a Proof at cut-the-knot

A sample application of Faltings ' theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to a < sup > n </ sup > + b < sup > n </ sup > = c < sup > n </ sup >, since for such n the curve x < sup > n </ sup > + y < sup > n </ sup > = 1 has genus greater than 1.

A mathematical analogy of self-referential statements lies at the core of Gödel's Incompleteness Theorem.

* A Proof of the Pythagorean Theorem From Heron's Formula at cut-the-knot

* James Gregory's Euclidean Proof of the Fundamental Theorem of Calculus at Convergence