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# 数学代写|数论代写Number Theory代考|Gaps Both Large and Small

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## 数学代写|数论代写Number Theory代考|Gaps Both Large and Small

It is not at all hard to find arbitrarily large gaps in the sequence of prime numbers – that is, to find arbitrarily long strings of consecutive composite numbers. In fact we can produce $n$ consecutive composite numbers by using the factorial function as follows:
$$(n+1) !+2,(n+1) !+3,(n+1) !+4,(n+1) !+5, \ldots,(n+1) !+(n+1) .$$
So, for example, for $n=10$, we get $11 !+2=39916802$, and we know that the ten consecutive numbers
$39916802,39916803,39916804,39916805, \ldots, 39916811$
will all be composite because the first number has to be divisible by 2 , the second by 3 , the third by 4 , and so on, until the last number is divisible by 11. Or, more ambitiously, if we want a million consecutive composite numbers, we let $n=1000000$, and produce
$$1000001 !+2,1000001 !+3, \ldots, 1000001 !+1000001$$
This certainly supports the idea that, as we expect, the primes do get farther apart as we get further out in the sequence of integers.

On the other hand, and this is really quite surprising, it also seems that extremely small gaps between primes continue to mysteriously persist as we get further out in the sequence of integers. In fact, other than the unique gap between the even prime 2 and the first odd prime 3 , the smallest possible gap, a gap of size 2-such as the gap between 17 and 19 , or 59 and 61 , or 1000000000061 and $1000000000063-$ seems to continue to persist no matter how far out we go in the sequence of primes.

In spite of the fact that the prime number theorem tells us that the primes get more and more rare as we go further and further out in the sequence of integers, every once in a while we come across a pair of primes that are as close to one another as they can possibly be. Such a pair of primes-that is, two primes $p$ and $p+2-$ are called twin primes.

## 数学代写|数论代写Number Theory代考|The Twin Prime Conjecture

One of the great joys of number theory as a subject is that it has provided many interesting questions that are simple to pose and yet remain unanswered after many years, in spite of enormous effort by mathematicians throughout the world. The twin prime conjecture is among the most famous unsolved problems in mathematics and proposes an answer to one such question: Are there infinitely many primes $p$ such that both $p$ and $p+2$ are prime? This conjecture optimistically claims that yes indeed the sequence of prime numbers never runs out of such twin primes.

Overwhelming data support the twin prime conjecture. For example, to find all 35 pairs of twin primes below 1000 , and all 8169 pairs below 1000000 , is relatively straightforward, and at this time all twin primes below 1000000000000000000 have been found! Here is a pair of twin primes with more than a hundred thousand digits each:
$65516468355 \cdot 2^{333333}-1$ and $65516468355 \cdot 2^{333333}+1$.
Yet a proof of the twin prime conjecture is nowhere in sight.
On the plus side, however, it was proved in 1966 that there are infinitely many pairs $p$ and $p+2$ such that $p$ is prime and $p+2$ has at most two prime factors. Problem 10.4 also provides strong support for the twin prime conjecture.

A result known as Brun’s theorem tells us something significant about the way in which twin primes thin out as we go further and further out in the sequence of integers. This theorem, proved in 1915 by the Norwegian mathematician Viggo Brun, says that the series consisting of the sum of the reciprocals of the twin primes converges; that is, the series
$$\left(\frac{1}{3}+\frac{1}{5}\right)+\left(\frac{1}{5}+\frac{1}{7}\right)+\left(\frac{1}{11}+\frac{1}{13}\right)+\left(\frac{1}{17}+\frac{1}{19}\right)+\cdots$$
converges. For a proof of Brun’s theorem see W. J. LeVeque, Fundamentals of Number Theory (New York: Dover, 1996).

It is natural to extend the idea of twin primes and to ask whether there are any prime triples, that is, any set of three consecutive odd numbers $n, n+2$, and $n+4$ that are all prime. Obviously, the numbers 3,5 , and 7 form such a triple. But, since for any three consecutive odd numbers one of the three numbers must be divisible by 3 , the set ${3,5,7}$ is the only possible such prime triple (see Problem 10.5).

Since looking for prime triples of the form $p, p+2$, and $p+4$ turns out to be not very interesting, it has instead become standard to call three numbers either of the form $p, p+2$, and $p+6$ or of the form $p, p+4$, and $p+6$ prime triplets if all three numbers are prime. The motive for this definition is that-except for the set ${3,5,7}$-this is as close as three odd primes can be. Some examples of prime triplets are ${5,7,11},{7,11,13},{17,19,23}$, and ${37,41,43}$. The most obvious question about prime triplets is whether there are infinitely many. The largest know prime triplet was found in 2012 and consists of the three numbers $81505264551807 \cdot 2^{33444}-1,81505264551807 \cdot 2^{33444}+1$, and $81505264551807 \cdot 2^{33444}+5$, each having 10082 digits.

## 数学代写|数论代写Number Theory代考|Gaps Both Large and Small

$$(n+1) !+2,(n+1) !+3,(n+1) !+4,(n+1) !+5, \ldots,(n+1) !+(n+1) .$$

$39916802,39916803,39916804,39916805, \ldots, 39916811$

$$1000001 !+2,1000001 !+3, \ldots, 1000001 !+1000001$$

## 数学代写|数论代写Number Theory代考|The Twin Prime Conjecture

$65516468355 \cdot 2^{333333}-1$和$65516468355 \cdot 2^{333333}+1$。

$$\left(\frac{1}{3}+\frac{1}{5}\right)+\left(\frac{1}{5}+\frac{1}{7}\right)+\left(\frac{1}{11}+\frac{1}{13}\right)+\left(\frac{1}{17}+\frac{1}{19}\right)+\cdots$$

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