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.

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

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.

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. 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,.

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.

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

Theorem 12.

Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).

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.

Theorem 13.

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: K is not a computable function.

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

Wiles discovered Fermat's Last Theorem on his way home from school when he was 10 years old.

* Lawrence C. Paulson of the University of Cambridge, work on higher-order logic system, co-developer of the Isabelle Theorem Prover

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

* Theorem Let X be a normed space.

* Theorem Every reflexive normed space is a Banach space.

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

George Boolos ( 1989 ) built on a formalized version of Berry's paradox to prove Gödel's Incompleteness Theorem in a new and much simpler way.

* Weisstein, Eric W. " Second Fundamental Theorem of Calculus.

: Theorem XXX: " The following classes of partial functions are coextensive, i. e. have the same members: ( a ) the partial recursive functions, ( b ) the computable functions.

: 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.

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.

The main difficulty in verifying Perelman's proof of the Geometrization conjecture was a critical use of his Theorem 7. 4 in the preprint ' Ricci Flow with surgery on three-manifolds '.

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

Also containing proofs of Perelman's Theorem 7. 4, there is a paper of Morgan and Tian, another paper of Kleiner and Lott, and a paper by Cao and Ge.

* ( Theorem 34. 4 )

Theorem: A group decision function with an odd number of voters meets conditions 1, 2, 3, and 4 if and only if it is the simple majority method.

* Old axiom II. 4 ( Pasch's Theorem ) is renamed as Theorem 4 and moved.

* A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. xii + 544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0 ( see Theorem 4. 5 )

Theorem V. 4. 5, p. 156

The equivalence of ( 4 ) and ( 5 ) ( Property ( FH )) is the Delorme-Guichardet Theorem.

The fact that ( 5 ) implies ( 4 ) requires us to assume that G is σ-compact ( and locally compact ) ( Bekka et al., Theorem 2. 12. 4 ).

( This subject reappears as Proposition 4, Theorem 4 in the Principia, and the corollaries here reappear also.

Theorem 4 shows that with a centripetal force inversely proportional to the square of the radius vector, the time of revolution of a body in an elliptical orbit with a given major axis is the same as it would be for the body in a circular orbit with the same diameter as that major axis.

* The Murderous Maths of Everything, ISBN 1-407-10367-9 ( prime numbers, Sieve of Eratosthenes, Pythagoras ' Theorem, triangle numbers, square numbers, the International Date Line, geometry, geometric constructions, topology, Mobius strips, curves ( conic sections and cycloids Golomb Rulers, 4 dimensional " Tic Tac Toe ", The Golden Ratio, Fibonacci sequence, Logarithmic spirals, musical ratios, Theorems ( including Ham sandwich theorem and Fixed point theorem ), probability ( cards, dice etc.

* Rewrite-Based Equational Theorem Proving with Selection and Simplification, Leo Bachmair and Harald Ganzinger, Journal of Logic and Computation 3 ( 4 ), 1994.

** Theorem 4: Time spent communicating and instance of symmetric exchanges are inversely related.