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. 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.

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.

Theorem 3: Necessarily, the property of being God-like is exemplified.

Pavel Samuilovich Urysohn, Pavel Uryson () ( February 3, 1898, Odessa – August 17, 1924, Batz-sur-Mer ) was a Jewish mathematician who is best known for his contributions in the theory of dimension, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology.

** Andrew Wiles wins worldwide fame after presenting his proof of Fermat's Last Theorem, a problem that had been unsolved for more than 3 centuries.

In projective spaces of dimension 3 or greater there is a construction that allows one to prove Desargues ' Theorem.

The result may be found in Coddington & Levinson ( 1955, Theorem 1. 3 ) or Robinson ( 2001, Theorem 2. 6 ).

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.

In chapter 3, the Prime Number Theorem ( PNT ) is introduced.

By Theorem 1. 3. 1 of Bekka et al., G is compactly generated.

A generalized form of Zariski Main Theorem is the following :< ref > EGA IV < sub > 3 </ sub >, Théorème 8. 12. 6 .</ ref > Suppose Y is quasi-compact and quasi-separated.

Theorem 3 now evaluates the centripetal force in a non-circular orbit, using another geometrical limit argument, involving ratios of vanishingly small line-segments.

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

** Theorem 3: Time spent in interaction and questions posed are inversely related.

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