* 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 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:

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

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

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p < 100, but according to Andrea del Centina, “ she had actually shown that it holds for every exponent p < 197 .” L. E. Dickson later used Germain's theorem to prove Fermat's Last Theorem for odd primes less than 1700.

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.

* Dylan G. L. Allegretti, Simplicial Sets and van Kampen's Theorem ( Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets ).

Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.

Though she never scaled the heights of her contemporaries Sophia Loren and Gina Lollobrigida, Mangano remained a favorite star between the 1950s and 1970s, appearing in Anna ( Alberto Lattuada, 1951 ), The Gold of Naples ( L ' oro di Napoli, Vittorio De Sica, 1954 ), Mambo ( Robert Rossen, 1955 ), Theorem ( Teorema, Pier Paolo Pasolini, 1968 ), Death in Venice ( Morte a Venezia, Luchino Visconti, 1971 ), and The Scientific Cardplayer ( 1972 ).

: 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 >.

* Kahneman, D., Knetsch, J. L., Thaler, R. H. " Experimental Tests of the Endowment Effect and the Coase Theorem " ( 1990 ) Journal of Political Economy, 98 ( 6 ), 1325-1348.

* L. Le Cam ( 1986 ) The Central Limit Theorem Around 1935, Statistical Science, Vol.

many small primes p, and then reconstructing B < sub > n </ sub > via the Chinese Remainder Theorem.

By Fermat's Little Theorem, 2 < sup >( q − 1 )</ sup > ≡ 1 ( mod q ).

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

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 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.

* Fermat's Last Theorem, about integer solutions to a < sup > n </ sup > + b < sup > n </ sup > = c < sup > n </ sup >

:: Theorem ( Lefschetz theorem on ( 1, 1 )- classes ) Any element of H < sup > 2 </ sup >( X, Z ) ∩ H < sup > 1, 1 </ sup >( X ) is the cohomology class of a divisor on X.

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 ).

The reader will find it helpful to think of the special case when the primes are of degree 1, and even more particularly, to think of the proof of Theorem 10, a special case of this theorem.

The identity of is unique by Theorem 1. 4 below.

The inverse of is unique by Theorem 1. 5 below.

Theorem 1. 2:

Theorem 1. 2a:

Theorem 1. 4: The identity element of a group is unique.

Theorem 1. 5: The inverse of each element in is unique.

Therefore both and are inverses of By Theorem 1. 5,

The conclusion follows from Theorem 1. 4.

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

Theorem 1: If a property is positive, then it is consistent, i. e., possibly exemplified.

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.

By the Chinese Remainder Theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the only idempotents will be 0 and 1.