There is no formal distinction between a lemma and a theorem, only one of intention – see Theorem # Terminology.

:: Boone-Rogers Theorem: There is no uniform partial algorithm which solves the word problem in all finitely presented groups with solvable word problem.

( There is a fundamental theorem holding in every finite group, usually called Fermat's little Theorem because Fermat was the first to have proved a very special part of it.

There are now several different proofs of Perelman's Theorem 7. 4, or variants of it which are sufficient to prove geometrization.

There is a Computational Representation Theorem in the Actor model for systems which are closed in the sense that they do not receive communications from outside.

There are two aphorisms that permit observers to calculate Variety ; four Principles of Organization ; the Recursive System Theorem ; three Axioms of Management and a Law of Cohesion.

: Theorem: There is no greatest cardinal number.

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.

By Theorem 10, D is a diagonalizable operator which we shall call the diagonalizable part of T.

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 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

Theorem: K is not a computable function.

He is most famous for proving Fermat's Last Theorem.

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

* Theorem Every reflexive normed space is a Banach space.

: 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.

One particularly important physical result concerning conservation laws is Noether's Theorem, which states that there is a one-to-one correspondence between conservation laws and differentiable symmetries of physical systems.

: 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 ).

Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations — including the " Last Theorem "— were printed in this version.

( This is the Fundamental Theorem of Equivalence Relations, mentioned above );

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

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

The local density of points in such systems obeys Liouville's Theorem, and so can be taken as constant.

The Hardy – Weinberg principle ( also known by a variety of names: HWP, Hardy – Weinberg equilibrium model, HWE, Hardy – Weinberg Theorem, or Hardy – Weinberg law ) states that both allele and genotype frequencies in a population remain constant — that is, they are in equilibrium — from generation to generation unless specific disturbing influences are introduced.

Liouville's Theorem shows that, for conserved classical systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble ( i. e., the total or convective time derivative is zero ).

The Index Theorem states that this analytical index is constant as you vary the elliptic operator smoothly.

The proofs of Nevanlinna and Ahlfors indicate that the constant 2 in the Second Fundamental Theorem is related to the Euler characteristic of the Riemann sphere.

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:

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 >

Pappus of Alexandria (; Greek ) ( c. 290 – c. 350 ) was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection ( c. 340 ), and for Pappus's Theorem in projective geometry.

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:

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.

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

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

Fermat's Last Theorem is a particularly well-known example of such a theorem.

Several theories attempt to explain and quantify the relationship between animals and their food, such as Kleiber's law, Holling's disk equation and Marginal Value 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.

Apart from his research work, Yudkowsky has written explanations of various philosophical topics in non-academic language, particularly on rationality, such as " An Intuitive Explanation of Bayes ' Theorem ".

It explains to a general audience various matters relating to the Lucasian professor's work, such as Gödel's Incompleteness Theorem and P-branes ( part of superstring theory in quantum mechanics ).

The unclear status of the Second Theorem was noted for several decades by logicians such as Georg Kreisel and Leon Henkin, who asked whether the formal sentence expressing " This sentence is provable " ( as opposed to the Gödel sentence, " This sentence is not provable ") was provable and hence true.

In its general form, the Löwenheim – Skolem Theorem states that for every signature σ, every infinite σ-structure M and every infinite cardinal number κ ≥ | σ | there is a σ-structure N such that | N |

They used graphics to represent their mathematical theories such as the Circle Theorem and the Pythagorean theorem.

Forsythe directed the Frankfurt Ballet ( Ballett Frankfurt ) from 1984 until 2004, choreographing such seminal pieces such as Artifact ( 1984 ), Die Befragung des Robert Scott ( 1986 ) Impressing the Czar ( 1988 ), Limb ’ s Theorem ( 1990 ), The Loss of Small Detail ( 1991 ), ALIE /< u > N A ( C ) TION </ u > ( 1992 ), Eidos: Telos ( 1995 ), Endless House ( 1999 ), Kammer / Kammer ( 2000 ), and Decreation ( 2003 ).

However, as there are many significant results in mathematics that make use of self-reference ( such as Gödel's Incompleteness Theorem ), the paradox illustrates some of the power of self-reference, and thus touches on serious issues in many fields of study.

LoF claimed that certain well-known mathematical conjectures of very long standing, such as the Four Color Theorem, Fermat's Last Theorem, and the Goldbach conjecture, are provable using extensions of the pa. Spencer-Brown eventually circulated a purported proof of the Four Color Theorem, but it met with skepticism.

This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular.

In a September 1904 lecture in St. Louis named The Principles of Mathematical Physics, Poincaré draw some consequences from Lorentz's theory and defined ( in modification of Galileo's Relativity Principle and Lorentz's Theorem of Corresponding States ) the following principle: " The Principle of Relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion.