Theorem: There is a constant c such that

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.

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.

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

concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he had found " a truly marvelous proof of this proposition ," now referred to as Fermat's Last Theorem.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

This theorem was established by John von Neumann, who is quoted as saying " As far as I can see, there could be no theory of games … without that theorem … I thought there was nothing worth publishing until the Minimax Theorem was proved ".

Using Rogers ' characterization of acceptable programming systems, Rice's Theorem may essentially be generalized from Turing machines to most computer programming languages: there exists no automatic method that decides with generality non-trivial questions on the black-box behavior of computer programs.

However, the Coase Theorem is difficult to implement because Coase does not offer a negotiation method. Additionally, firms could potentially bribe each other since there is little to no government interaction under the Coase Theorem.

In social choice theory, Arrow ’ s impossibility theorem, the General Possibility Theorem, or Arrow ’ s paradox, states that, when voters have three or more distinct alternatives ( options ), no rank order voting system can convert the ranked preferences of individuals into a community-wide ( complete and transitive ) ranking while also meeting a specific set of criteria.

In 1847 Gabriel Lamé announced a solution of Fermat's Last Theorem for all -- i. e., that the Fermat equation has no solutions in nonzero integers, but it turned out that his solution hinged on the assumption that the cyclotomic ring is a UFD.

* Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group ( Mordell's Theorem, later generalized to the Mordell – Weil theorem ).

Gödel's First Incompleteness Theorem then tells us that there are certain consistent bodies of propositions with no recursive axiomatization.

Similarly, mathematicians often describe Fermat's Last Theorem as asserting that there are no nontrivial integer solutions to the equation when n is greater than 2.

Moreover, as long as the polynomial factors at each stage are relatively prime ( which for polynomials means that they have no common roots ), one can construct a dual algorithm by reversing the process with the Chinese Remainder Theorem.

Bayes ' Theorem shows that the probability will never reach exactly 0 or 100 % ( no absolute certainty in either direction ), but it can still get very close to either extreme.

In the case where there is at least some wind, the Hairy Ball Theorem dictates that at all times there must be at least one point on a planet with no wind at all and therefore a tuft.

In axial symmetry, he considered general equilibrium for distributed currents and concluded under the Virial Theorem that if there were no gravitation, a bounded equilibrium configuration could exist only in the presence of an azimuthal current.

It was these equations which inspired Pierre de Fermat to propose Fermat's Last Theorem, scrawled in the margins of Fermat's copy of ' Arithmetica ', which states that the equation, where,, and are non-zero integers, has no solution with greater than 2.

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.

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

Theorem 1 demonstrates that where an orbiting body is subject only to a centripetal force, it follows that a radius vector, drawn from the body to the attracting center, sweeps out equal areas in equal times ( no matter how the centripetal force varies with distance ).

* 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