* James ` s Theorem For a Banach space the following two properties are equivalent:

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.

For larger values of n, Fermat's Last Theorem states there are no positive integer solutions ( x, y, z ).</ td >

Theorem 1. 6: For all elements in a group.

Theorem 1. 7: For all elements and in group,.

Theorem 1. 8: For all elements in a group, then.

Theorem 1. 3: For all elements in a group, there exists a unique such that, namely.

For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.

Theorem ( Modified Schwarz inequality for 2-positive maps ) For a 2-positive map Φ between C *- algebras, for all a, b in its domain,

For Theorem I the data are generated in full range, while in Theorem II data is only generated when it surpasses a certain threshold, called Peak Over Threshold models ( POT ).

For any flow, you can write the equations of the flow in terms of vorticity rather than velocity by simply taking the curl of the flow equations that are framed in terms of velocity ( may have to apply the 2nd Fundamental Theorem of Calculus to do this rigorously ).

For example Newton wrote to Oldenberg in 1676 " I believe M. Leibnitz will not dislike the Theorem towards the beginning of my letter pag.

For dimension 2, there is a rich structure in virtue of the absence of Desargues ' Theorem.

For example, when students are discovering the formula for the Pythagorean Theorem in math class, the teacher may identify hints or cues to help the student reach an even higher level of thinking.

( For example, the means of sufficiently large samples drawn from practically any underlying distribution whose variance exists are normally distributed, according to the Central Limit Theorem.

</ li >< li > Pythagorean Theorem: For and meeting on orthogonal lines at ( )

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 For a bounded operator T, σ < sub > r </ sub >( T ) ⊂ σ < sub > p </ sub >( T *) ⊂ σ < sub > r </ sub >( T ) ∪ σ < sub > p </ sub >( T ).

His Ph. D. dissertation, written under the direction of Richard Courant and Bernard Friedman at New York University, was on the topic " The Bridge Theorem For Minimal Surfaces.

: Theorem 1: For any pair of algorithms a < sub > 1 </ sub > and a < sub > 2 </ sub >

For such objects, the integral may be taken over the entire surface () by taking the absolute value of the integrand ( so that the " top " and " bottom " of the object do not subtract away, as would be required by the Divergence Theorem applied to the constant vector field ) and dividing by two:

For a proof of the first part of the Latin square property, see the Wikipedia page on elementary group theory ( Theorem 1. 3 ).

To see the equivalence, note first that if Theorem 1 holds, and φ is not satisfiable in any structure, then ¬ φ is valid in all structures and therefore provable, thus φ is refutable and Theorem 2 holds.

If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬ φ is not satisfiable in any structure and therefore refutable ; then ¬¬ φ is provable and then so is φ, thus Theorem 1 holds.

In the General Possibility Theorem, Kenneth Arrow argues that if a legislative consensus can be reached through a simple majority, then minimum conditions must be satisfied, and these conditions must provide a superior ranking to any subset of alternative votes ( Arrow 1963 ).

Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π ( x ) were strong enough for him to prove Bertrand's postulate that there exists a prime number between n and 2n for any integer n ≥ 2.

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 ):

Strange loops take form in human consciousness as the complexity of active symbols in the brain inevitably lead to the same kind of self-reference which Gödel proved was inherent in any complex logical or arithmetical system in his Incompleteness Theorem.

The Court of Appeal upheld the conviction, but it also gave the opinion that " To introduce Bayes ' Theorem, or any similar method, into a criminal trial plunges the jury into inappropriate and unnecessary realms of theory and complexity, deflecting them from their proper task.

* Gödel's Incompleteness Theorem: any logical model of reality is incomplete ( and possibly inconsistent ) and must be continuously refined / adapted in the face of new observations.

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.

He did this by attempting to show that any counterexample to Fermat's Last Theorem would give rise to a non-modular elliptic curve.

Using Wilson's Theorem, for any odd prime we can rearrange the left hand side of

This was derived from the Second Law ( any of the two, Clausius ' or Lord Kelvin's statement can be used since they are equivalent ) and using the Clausius ' Theorem ( see Kerson Huang ISBN-13: 978-0471815181 ).

Good on what is now called the prime-factor FFT algorithm ( PFA ); although Good's algorithm was initially mistakenly thought to be equivalent to the Cooley – Tukey algorithm, it was quickly realized that PFA is a quite different algorithm ( only working for sizes that have relatively prime factors and relying on the Chinese Remainder Theorem, unlike the support for any composite size in Cooley – Tukey ).

An extension of the halting problem is called Rice's Theorem, which states that it is undecidable ( in general ) whether a given language possesses any specific nontrivial property.

In projective geometry, Pascal's theorem ( aka Hexagrammum Mysticum Theorem ) states that if an arbitrary six points are chosen on a conic ( i. e., ellipse, parabola or hyperbola ) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon ( extended if necessary ) meet in three points which lie on a straight line, called the Pascal line of the hexagon.

Christian List showed that May's Theorem for two alternatives is simply a special case for a more general theorem that show that approval voting is the only voting system that is anonymous, neutral, monotone, and positively responsive for any arbitrary number of alternatives.

Pareto-optimal solutions are not unique, and according to the Second Fundamental Theorem of Welfare Economics, a social planner can achieve any Pareto-optimal outcome by an appropriate redistribution of wealth by means of competitive market.

Bell's Theorem states that the predictions of quantum mechanics cannot be reproduced by any local hidden variable theory.

Hutchings's Theorem states that any of the prime numbers 5, 7, 11, 13, …, wins as a first move, but very little is known about the subsequent winning moves: these are the only winning openings known.

Green's Theorem states that for any H-class H of a semigroup S either ( i ) or ( ii ) and H is a subgroup of S. An important corollary is that the equivalence class H < sub > e </ sub >, where e is an idempotent, is a subgroup of S ( its identity is e, and all elements have inverses ), and indeed is the largest subgroup of S containing e. No H-class can contain more than one idempotent, thus H is idempotent separating.