It is now relatively easy to check that the PNT is equivalent to the claim that.

Finally, we can conclude that the PNT is " morally " true.

The United States Naval Observatory ( USNO ) is one of the oldest scientific agencies in the United States, with a primary mission to produce Positioning, Navigation, and Timing ( PNT ) for the U. S. Navy and the U. S. Department of Defense.

The prime number theorem ( PNT ) implies that the number of primes up to x is roughly x / log ( x ), so if we replace x with 2x then we see the number of primes up to 2x is asymptotically twice the number of primes up to x ( the terms log ( 2x ) and log ( x ) are asymptotically equivalent ).

So Bertrand's postulate is comparatively weaker than the PNT.

Again we expect that there will be not just one but many primes between n < sup > 2 </ sup > and ( n + 1 )< sup > 2 </ sup >, but in this case the PNT doesn't help: the number of primes up to x < sup > 2 </ sup > is asymptotic to x < sup > 2 </ sup >/ log ( x < sup > 2 </ sup >) while the number of primes up to ( x + 1 )< sup > 2 </ sup > is asymptotic to ( x + 1 )< sup > 2 </ sup >/ log (( x + 1 )< sup > 2 </ sup >), which is asymptotic to the estimate on primes up to x < sup > 2 </ sup >.

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

The Positioning, Navigation, and Timing ( PNT ) program is responsible for coordinating and developing PNT technology, PNT policy, and spectrum management.

PNT is crucial for planning and tracking shipments, mapping routes, and other transportation functions, and is increasingly used for safety functions.

In number theory, the prime number theorem ( PNT ) describes the asymptotic distribution of the prime numbers.

So unlike the previous case of x and 2x we don't get a proof of Legendre's conjecture even for all large n. Error estimates on the PNT are not ( indeed, cannot be ) sufficient to prove the existence of even one prime in this interval.

The name Paramount Network Television ( PNT ) was also used for the first time ever ( Viacom Productions would also become a division of PNT ).

In 2004, the IGEB was superseded by the National Executive Committee for Space-Based Positioning, Navigation and Timing ( PNT ), established by presidential order.

* Archive of the IGEB at the National Executive Committee for Space-Based Positioning, Navigation, and Timing ( PNT )

Like most proofs of the PNT, it starts out by reformulating the problem in terms of a less intuitive, but better-behaved, prime-counting function.

In an attempt to reduce the impact of these threats, Petra National Trust ( PNT ) was established in 1989.

All modes of transportation rely on the Global Positioning System ( GPS ) and other PNT capabilities to provide safe and reliable operation of transportation systems.

The Office of Positioning, Navigation, and Timing represents all Federal civil departments and agencies in the development, acquisition, management, and operations of GPS, as well as providing civil PNT system analysis including requirements and architecture development and performance monitoring.

They are huge areas which have been swept by winds for so many centuries that there is no soil left, but only deep bare ridges fifty or sixty yards apart with ravines between them thirty or forty feet deep and the only thing that moves is a scuttling layer of sand.

It consists of fragmentary personal revelations, such as `` The Spark '': `` There is a spark dwells deep within my soul.

It is testimony to the deep respect in which Mr. Eisenhower was held by members of all parties that the moral considerations raised by his approach to the matter were not explicitly to be broached.

Mann understood better than most men the incest comedy at the center of the myth and the psychological truth in which dread is shown as the other face as longing was for him just the kind of deep and complicated joke he liked to tell.

The road, a comparatively new one, is very good, winding along inlets, coves, and bays of deep and brilliant blue.

The religious quest is often intense and deep, and there are students on every campus who are seriously wrestling with the most profound questions of meaning and value.

Now Richards, of course, is known as a deep thinker as baseball managers go.

It is a matter of deep personal satisfaction for me to add my voice to the great and distinguished chorus of my colleagues in this paean of praise, respect, and affection for Speaker Sam Rayburn.

It is, I insist, hard to define the Rayburn contribution to our political civilization because it is so massive and so widespread and so complicated, and because it goes so deep.

While clay is still pressed in mold, press three equally spaced holes 1/4'' '' deep, using pencil eraser, in bottom of clay to allow for proper drying and firing.

I took a deep breath and an even deeper swallow of my drink, and said, `` I admit that going back to Ralph Waldo Emerson for humor is like going to a modern musical comedy for music and comedy ''.

The deep water is used by many people, but it is always clean, for the washing is done outside.

The history of the U.S. potato chip industry is that many of today's successful companies got started during the deep depression days.

`` The white colonnaded, cedar-roofed Southern mansion is directly traceable via the grey and buff stone of grey-skied England to the golden stucco of one particular part of the blue South, the Palladian orbit stretching out from Vicenza: the old mind of Andrea Palladio still smiles from behind many an old rocking chair on a Southern porch, the deep friezes of his architectonic music rise firm above the shallower freeze in the kitchen, his feeling for light and shade brings a glitter from a tall mint julep, his sense of columns framing the warm velvet night has brought together a million couple of mating lips ''.

The statue is the " thing in itself ", and his slender face with the deep eyes express an intellectual eternity.

The depth of water at the apex of the ridge is less than in most places, while the bottom of the ridge is three times as deep.

The deep ocean floor is thought to be fairly flat with occasional deeps, abyssal plains, trenches, seamounts, basins, plateaus, canyons, and some guyots.

With the development of fast Internet in the last part of the 20th century along with advances in computer controlled telescope mounts and CCD cameras ' Remote Telescope ' astronomy is now a viable means for amateur astronomers not aligned with major telescope facilities to partake in research and deep sky imaging.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.