If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If Af is the null space of Af, then Theorem 12 says that Af.

* Theorem If X is a normed space, then X ′ is a Banach space.

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

Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.

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.

If we also include the German speaking Vienna, during the Weimar years Mathematician Kurt Gödel published his groundbreaking Incompleteness Theorem.

If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal.

Theorem: If R is a filtered ring whose associated graded ring gr ( R ) is a domain, then R itself is a domain.

If we consider measuring distance based on Pythagoras ' Theorem then it is clear that we shall be adding quantities measured in different units, and so this leads to meaningless results.

* ( Kronecker's Theorem ) If p is an irreducible monic integer polynomial with, then either p ( z )=

Theorem ( Gregoire de Saint-Vincent 1647 ) If bc

If now diameter AF is drawn bisecting DC so that DF and CF are sides c of an inscribed decagon, Ptolemy's Theorem can again be applied – this time to cyclic quadrilateral ADFC with diameter d as one of its diagonals:

:: If, then by Fermat's Little Theorem each of the numbers is congruent to one modulo.

: Theorem: If S is any set then S cannot contain elements of all cardinalities.

Theorem: K is not a computable function.

Major industrial companies in Mysore include Bharat Earth Movers, J. K. Tyres, Wipro, Falcon Tyres, Larsen & Toubro, Theorem India and Infosys.

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.

* 1873-Biophysicist Hermann von Helmholtz develops a mathematical law of bird flight in Uber ein Theorem, geometrisch Ohnliche Bewegungen flussiger Korper betreffend, nebst Anwendung auf das Problem, Luftballons zu lenken ( Monatsbericht d. K. Akad.

Turán's best-known result in this area is Turán's Graph Theorem, that gives an upper bound on the number of edges in a graph that does not contain the complete graph K < sub > r </ sub > as a subgraph.

* The integral of the Gaussian curvature K of a 2-dimensional Riemannian manifold ( M, g ) is invariant under changes of the Riemannian metric g. This is the Gauss-Bonnet Theorem.

Theorem: if K is an algebraically closed field of characteristic zero, then the field of Puiseux series over K is the algebraic closure of the field of formal Laurent series over K.

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

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.

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 >

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.

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.

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 >.

The Relative Hurewicz Theorem states that if each of X, A are connected and the pair ( X, A ) is ( n − 1 )- connected then H < sub > k </ sub >( X, A ) = 0 for k < n and H < sub > n </ sub >( X, A ) is obtained from π < sub > n </ sub >( X, A ) by factoring out the action of π < sub > 1 </ sub >( A ).

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

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.

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

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

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

The conclusion follows from Theorem 1. 4.

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.

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

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.

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.