Its name derives from the fact that period length is chosen to be a Mersenne prime.

In fact, the base-2 repunits are the well-respected Mersenne numbers M < sub > n </ sub > = 2 < sup > n </ sup > − 1.

Although Cataldi also claimed that p = 23, 29, 31 and 37 all also generate Mersenne primes ( and perfect numbers ), his text's clear demonstration shows that he had genuinely established the fact through p = 19.

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

And so far, every one thus tested has in fact been the sum of two primes.

Since this quantity goes to infinity as n increases, we expect that every large even integer has not just one representation as the sum of two primes, but in fact has very many such representations.

This result was subsequently enhanced by many authors ; currently, the best known result is due to Olivier Ramaré, who in 1995 showed that every even number n ≥ 4 is in fact the sum of at most six primes.

In fact, resolving the weak Goldbach conjecture will also directly imply that every even number n ≥ 4 is the sum of at most four primes .< ref >

He proved also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i. e.

In fact, although Gauss also conjectured that there are infinitely many primes such that the ring of integers of is a PID, to this day we do not even know whether there are infinitely many number fields ( of arbitrary degree ) such that is a PID!

Olivier Ramaré in 1995 showed that every even number n ≥ 4 is in fact the sum of at most six primes.

Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions, and secondly on his proof that this can be done if the odd prime factors of n are distinct Fermat primes, which are of the form.

( In modern language, we now say that the existence of infinitely many primes is reflected by the fact that the Riemann zeta function has a simple pole at s

These methods rely on the fact that finding two large primes and multiplying them together ( resulting in a semiprime ) is computationally simple, whereas finding the original factors appears to be difficult.

In fact only primes p ramifying in the extension of ℚ generated by a root of Q have any chance of working.

) In fact, Eisenstein polynomials are directly linked to totally ramified primes, as follows: if

In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes.

In fact, Euclid's proof did not assume that a finite set contains all primes that exist.

The statement of the more general Chebotarev theorem is in terms of the Frobenius element of a prime ( ideal ), which is in fact an associated conjugacy class C of elements of the Galois group G. If we fix C then the theorem says that asymptotically a proportion | C |/| G | of primes have associated Frobenius element as C. When G is abelian the classes of course each have size 1.

This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers B < sub > 2n </ sub > as the product of all primes p such that p − 1 divides 2n ; consequently the denominators are square-free and divisible by 6.

In fact, Euclid did not begin with the assumption that the set of all primes is finite.

In fact that statement is a little weaker, in that it disregards any initial partial sum of the series, in a way that may vary according to p. For the other primes the radius is non-zero.

In fact, if they assume the Elliott – Halberstam conjecture, then they can also show that primes within 16 of each other occur infinitely often, which is related to the twin prime conjecture.

And, as a matter of fact, Nicolas had slept in the park only part of one night, when he discovered that Munich's early mornings even in summer are laden with dew.

In fact, although we have dispelled the fear, we have not necessarily assured ourselves that there are no dangers.

In fact, insofar as science generates any fear, it stems not so much from scientific prowess and gadgets but from the fact that new unanswered questions arise, which, until they are understood, create uncertainty.

We are also struck by the fact that this story of a boy's love for his mother does not offend, while the incestuous love of the man, Paul Morel, sometimes repels.

we accord it its place there, and in Lawrence's treatment we are given the innocent fantasy of a child, in fact, the form in which oedipal love is expressed in childhood.

There is probably some significance in the fact that two of the best incest stories I have encountered in recent years are burlesques of the incest myth.

Some of us might be inclined to argue, in fact, that an independence of mind and action and an intolerance of regimentation, either mental or physical, are particularly Southern traits.

There is evidence to suggest, in fact, that many authors of the humorous sketches were prompted to write them -- or to make them as indelicate as they are -- by way of protesting against the artificial refinements which had come to dominate the polite letters of the South.

This finding is consistent also with the fact that student leaders are more likely to be supporters of the values implicit in civil liberties than the other students.

In fact, if judgments are to be rendered upon the soundness of his historicism, they must be based on scanty evidence.

That fact is very clearly illustrated in the case of the many present-day intellectuals who were Communists or near-Communists in their youth and are now so extremely conservative ( or reactionary, as many would say ) that they can define no important political conviction that does not seem so far from even a centrist position as to make the distinction between Mr. Nixon and Mr. Khrushchev for them hardly worth noting.

It is a fact of life that magazines are edited by groups: they have to be or they wouldn't be published at all.

But if anything can bring home to Mr. Khrushchev the idea that he will not really get much enjoyment from watching this Braddock-against-the-Indians contest, it will probably be the fact that SEATO forces are ready to attempt it -- plus the fact that Moscow has something to lose from closing off disarmament and other bigger negotiations with Washington.

There is a trend today to bemoan the fact that Americans are too `` soft ''.

the rub arises from the fact that teachers are usually paid on the basis of time served rather than quality.

This caution has been particularly noticeable in a tendency of retailers and distributors to shift the inventory burden back on the supplier, and the fact stocks at retail are low in many lines has escaped attention because of the presence of higher stocks at the manufacturing level.

But Schnabel was a great teacher in addition to being a great performer, and the fact that four of the ten versions I listened to are by Schnabel pupils ( Clifford Curzon, Frank Glazer, Adrian Aeschbacher, and Victor Babin ) also sheds light on the master's pedagogical skills.

Some of them are obvious, such as the fact that we associate recorded and live music with our responses and behavior in different types of environments and social settings.

Due to the fact that building codes and regulations vary so much throughout the country, the first thing to do is to find out what, if any, they are.

Some distribution costs are kept up by competitive pressure, some by the fact that the customers have come to expect certain niceties and flourishes.

Fundamental to the difficulty of creating the desired prestige is the fact that, in the business community, prestige and status are conferred in proportion to the authority that one man has over others and the extent of which he participates in the management functions ''.

Important as these differences are, they should not obscure the basic fact that by shifting the hypothalamic balance sufficiently to the parasympathetic side, we produce depressions, whereas a shift in the opposite direction causes excitatory effects and, eventually, maniclike changes.

Perhaps a clue to these and related problems lies in the fact that changes in the intensity of hypothalamic discharges which are associated with changes in its balance lead also to qualitative alterations in reactivity.