Sum Numbers and prime Numbers

Have you ever thought about: why will happen to is divided into 24 hours a day, the circumference of a circle is divided into 360 degrees?24 this number is actually a very interesting properties: it can be in the form of more divided into several equal parts completely.24 members present, for example, 2 = 12, 24 members present 3 = 8, 24 members present 4 = 6, and so on (you can write out all the rest of the rule of thirds!)This means that one day can be divided into two equal parts, each part 12 hours, day and night.8 hours for one shift uninterrupted work of the factory, every day can be precisely divided into three shifts.

This is why the circumference of a circle is divided into 360 degrees.If the circumference of a circle is divided into two, three, four, 10, 12, or thirty equal parts, each part of the degree were as an integer;Of course, we didn't mention other segmentation method of circular.In ancient times, in order to meet the demands of art, astronomy, and engineering and other fields, circumferential accuracy can be divided into several equal the size of the fan is very necessary.This work has great practical significance and the process in addition to compass and protractor, there is no other available tools¹.

Can be written as two of the smaller positive integers, is not equal to 1, the product of the positive integer, is calledSum Numbers.Equation, for example, 24 = 3 x 11 = 4 x 6 and 33 suggests that 24 and 33 are sum Numbers.Unable to break down in this way not a positive integer is calledPrime Numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23 and 29

Are prime Numbers.In fact, they are the top 10 primes (if want to check their own!.

Observation that a small column of prime Numbers are enough to some interesting findings.First of all, except for 2, all the prime Numbers are odd Numbers, this is because an even number can be divided exactly by 2, which makes the even Numbers except 2 are sum Numbers.Therefore, any two consecutive than the distance between the two large prime Numbers at least 2.In this a small column of prime Numbers, we can find a two consecutive prime, which bad for just 2 (3 and 5, for example, 17 and 19).Continuous difference of two prime Numbers can also is bigger, such as between 23 and 29 are six, and between 24, 25, 26, 27, 28 are sum Numbers.Another interesting finding is that in the first and the second group made up of 10 consecutive positive integers (that is 1-10 and 11-20) there are four prime, but in the third group (i.e. 21-30) only two prime Numbers.What does it mean?, with the increase of number of primes becomes more scarce?Anyone can say we will keep looking for, find infinitely many prime Numbers?

If at this time, what makes you feel excited, and want to continue to study the issue of prime number sequence and we just put forward, which means that your mind is like a mathematician.Stop!Don't rush to look down²!Get a pen, and paper, write down all the number from 1 to 100, then prime Numbers.See how many differ for 2 set of prime Numbers, see every 10 the number of how many consecutive prime Numbers.You can find any law?Or no more than 100 of these prime Numbers is chaotic?

Some myth and a theorem

Prime Numbers in the ancient times has attracted people's attention, and even been linked with the supernatural.Even today, in modern times, there are also many people give mysterious attributes to prime Numbers.Famous astronomer and author ⋅ Carl sagan in 1985 wrote a book called "contact", tells the story of an alien civilization trying to communicate in prime Numbers that serve as signals and human.This signal as an alien civilization based on the prime way to communicate ideas, still kept arousing people's imagination.

It is generally believed that in the age of Pythagoras, people have produced strong interest of prime Numbers.Pythagoras was the ancient Greek mathematician.His students of the Pythagorean school live in the 6th century b.c., is one part of a scientist, another part is the mystic.They didn't leave written records;Our understanding of them are all from word of mouth.In three hundred, after the third century BC, Alexander (now in Egypt) is the cultural center of the Greek world.You may have heard of drag le dense I live in Euclidean Alexander (Figure 1), because the Euclidean geometry is named after him.Euclidean geometry is professor in the school for over 2000 years.But the Euclidean also interested in number theory;"Element" in his works of 20, volume 9 proposition given 'prime Numbers there are an infinite number' 'thistheoremOne of the firstMathematical proof.

At this time to talk about a theorem and mathematical proofs of these concepts is appropriate.theorem是指用数学语言所作的陈述, 具有要么成立要么不成立的确定性。比方说, ''质数有无穷多个'' 这个定理断言在自然数序列（1, 2, 3, ……）中质数序列是无限长的。更确切地说, 这个定理是说如果我们写出一个有限的质数序列, 我们总能找到另一个质数不在原有的序列中。要证明这个定理, 仅仅对特定给定的有限质数序列, 指出有另一个不同的质数, 是不够的。例如, 如果我们对于前 10 个质数构成的序列, 指出 31 是质数但不在这个序列之中, 我们的确说明了这个序列不包含所有的质数。但也许把 31 加进来后我们就找到了所有的质数, 而没有别的了。我们需要做的, 也恰恰是欧几里得 2300 年前做的, 是给出一个令人信服的论证——为什么对于任何有限的质数序列, 不论它有多长, 都能找到不在其中的质数。在下一部分中我们会介绍欧几里得的证明, 而且不涉及太多细节, 免得读起来无聊。

Euclid 'prime Numbers have an infinite number of' proof

Infinitely many prime Numbers, in order to prove the existence of Euclid used his known another fundamental theorem, namely 'any natural number can be written as the product of primes', and this thesis easily proved right.If you choose a positive integer greater than 1, it is not a sum Numbers, it is itself a prime Numbers.Otherwise, it can be written as the product of two positive integers smaller.If the two smaller Numbers are prime Numbers, the number of the original has been divided into the product of two prime Numbers;If not, the more the number of decimal divided into smaller than it product and so on.In this process, the product continues to be a smaller number of replace the sum Numbers.Because it is impossible to points forever, so this process must be in one step stop, all of those quantities are not points, which means they are prime.For example, we divided 72 into prime factor of the product:

72 = 12 * 6 = 3 * 4 * 6 = 3 * 2 * 2 * 6 = 3 * 2 * 2 * 2 * 3

Based on the basic facts, we can explain the Euclidean unlimited beautiful proved on the prime number set.We will use the first ten consisting of a prime number sequence to demonstrate this idea, but please note that the same idea applies to any finite sequence of prime Numbers.We we multiply together all the number in the sequence, and the results combined with 1.Call the number in N.(in fact N what is not important, because this argument are applicable to any prime sequence).

N = (2 x 3 x 5 x 7 x 11 * 17 * 13 19 * 23 * 29) + 1

N and other natural number greater than 1, can be written as the product of prime Numbers.What number they are?Or what are the prime factors of N?We don't know, because we haven't done, but there is one thing we can be sure: they can be divided exactly by N.However, with our series 2, 3, 5, 7,..., 23, 29 any of prime Numbers in addition to N, will be more than 1.We assume that this is all the prime Numbers of sequence, only to find that they are not divisible N.So the prime factors of N is not in sequence, to be exact, there must be more than 29 new prime Numbers.

Erato, sieve method

Less than 100 all the prime Numbers you are looking for?In what way?Are you one by one to inspect each number, to see if it can be divided exactly by the smaller number?In this way, must be very time-consuming.Erato turney (Figure 1) is one of the greatest scholars in hellenistic times, living in Euclidean after decades.He served as chief librarian library of Alexandria.This is the first in the history of the library, is one of the biggest library of the ancient world.In addition to mathematics, he to astronomy, music and geography are interest in, he is at the first with incredible accuracy circumference of the earth.In addition, he put forward a kind of intelligent method to find all prime Numbers within a given range.Due to the idea of this method is the screen all the composite Numbers, we call it the Erato, sieve method.

We take less than 100 prime number sequence (I hope they are in front of you) as an example demonstrate Erato turney sieve method (Figure 2).Put 2 circle, because it was the first prime Numbers, and then erases all multiples of larger than 2 2, namely all coupling number.Continue to see a didn't erase the number 3, it also has, cannot be written as the product of a decimal, circle can take it, because it is a prime number.Similarly, the bigger than 3 3 multiples are erased.Please note that some of the number (such as 6) have been cleaned off, while others, such as 9) now is about to erase.Circle the number next wipe - 5, as before, wipe away all the larger than 5 multiples of 5, 10, 15 and 20 had been wiped, but 25 and 35 now just to wipe.To continue in the same way - to what time?In fact, from the number of near 10 (7) started to round do not need to go on.Try to think about why.At this moment, all didn't erase the number less than 100 are prime, and circle can be rest assured!

The frequency of the prime

The frequencies of prime number have how old?1, 000, 000 to 1, 001, 000 (between one million and one billion and one thousand) about how many prime Numbers?1, 000, 000, 000 to 1, 000, 001, 000 (billion to 1000001 thousand) and probably how many?Can we estimate one trillion (1, 000, 000, 000, 000) to one trillion and one thousand the number of prime Numbers?

Calculation shows that, with the increase of number of primes is becoming more and more scarce.However, if we could find an accurate theorem to accurately depict the rarity of them?Originally is a great mathematician Carl ⋅ friedrich ⋅ gauss in 1793, which is when he was 16 years old, withguessIn the form of this theorem is presented.In the 19th century mathematicians bernhard ⋅ Riemann (Figure 1) invented many tool of prime Numbers, the modern study of prime work produced great impact.However, the formal proof of the theorem is given only in 1896, and the distance it is put forward for a century.That same year, France's Jacques Hadamard delaVallee - Poussin and Belgium (Figure 1) the two mathematicians provides two unrelated to prove, that is really surprising.More interestingly, both are born around the Riemann died.They prove theorem to the importance of the name 'prime number theorem ".

The exact expression of prime number theorem and prove the detailed process need to use advanced mathematical tools, so we have not discussed here.But roughly speaking, the prime number theorem shows that the frequencies of x primes around about is inversely proportional to the x number of bits.In the example above, the length of 1000 around one million ' ' ' 'window (we refers to the interval between one million and one billion and one thousand) in the number of prime Numbers, than billion' window 'in the same big near 50% more than the number of prime Numbers, (the proportion to find, just happened to be in one million and the number of' 0 '), will be larger than one trillion near the same 'window' twice as many as the number of prime Numbers in (here, in one trillion and one million the number of '0' than is ").In fact, the computer calculates that there are 75 prime Numbers in the first window, with 49 in the second window, the third window, is refers to between one trillion and one trillion and one thousand, only 37.

This information can also be to use a chart to say (Figure 3).You can see in the range of x 100 or less, as well as in x 1 or less, 000 range, until x number of primes PI (x) how to change.Please note that we move right along the x axis, each new prime, curve will increase 1, so the image is a ladder-like (Figure 3. A).To see in small scale, it is difficult to found in the chart law - easy to prove that we can find any long range, one of the prime number all have no, this means that this part of the interval curve corresponding to the image will not go up.On the other hand, a famous hypothesis (see below) asserts that there are an infinite number of ' 'The twin prime number"', also is bad for the primes of 2;This is on the diagram shown as step width is 2.However, in a larger scale, image looks very smooth,Figure 3 b);The large scale of smooth curve is well showed the content of the prime number theorem.

Indeed, some in mathematics phenomenon on a scale not see law, but on a larger scale regularity () to the smoothness of the image, in other words, as the scale growth rule is more and more accurate.This is not uncommon in mathematics.And probability of the system, such as the model of the coin toss, is like this.Is impossible to predict a single coin toss is against, but as long as the two sides is symmetrical, coin as time increases, will face up half of the time.Surprisingly, prime number system is not random, but it still behaves in many ways seems to be selected at random.

Abstract: who wants to be a millionaire?

Number theory, including prime research, everywhere is the unsolved puzzle;For hundreds of years, even the most clever mind repeatedly challenge also failed to crack.Some of these open question mathematical proposition has not been proved, but we believe they are right;We call these haven't prove theorem 'guess' 'or' hypothesis'.We have already mentioned "' there are an infinite number of twin prime - sent to 2 prime conjecture of ' '.Another famous conjecture is the goldbach conjecture.It is said that every even number (except 2) can be written as two primes and.For example: 16 = 13 + 3, 54 = 47 + 7.If you can prove that any one of the two, you will go down³.

'Riemann hypothesis "is arguably one of the most famous unsolved mysteries in maths, it's proponents bernhard ⋅ Riemann we already mentioned.Riemann in 1859 published his only a paper studies prime Numbers, in which he proposed a conjecture, which estimates the PI (x) (until x number of prime Numbers) and the prime number theorem gives the approximate value of the difference between how far.In other words, ' ' ' 'error of prime number theorem, which is the difference between the actual number of prime Numbers and the approximate formula, what can we know?Clay foundation to the problem as one of the seven major mathematical problems, solution, they receive a bonus of $one million!If you don't have so far is interested, this award may bring you a motivation...

Why this problem is important?Who is interested in?mathematics家评判一个数学问题的价值时, 主要基于问题的难度和问题本质中的美感。以这两条作标准, 质数的得分都很高。此外, 质数也有实用价值。过去几十年中的研究已经揭示出质数在加密（研究给机密信息编码的学科）中重要的作用。我们此前已提到卡尔 ⋅ 萨根的那部讲述外星文化用质数与人类进行联络的科幻著作, 但是, 在一个更热的领域——加密传输中, 质数被应用于民事及军事。这当然就不是科幻了。我们从 ATM 机取款用的是借记卡, 而我们和 ATM 机之间的信息传输都是加密的。和许多其它密码系统一样, 几乎所有借记卡用的都是 RSA（得名于发明者 Rivest, Shamir 和 Adleman）密码系统, 它基于的正是质数的性质。

Prime story is still full of mystery, stories about their unfinished...

The vocabulary

Composite (Composite Number):writeTwo small positive integer can be written as the product of the positive integers, for example, 24 are sum Numbers, because 24 = 3 x 8.

Prime Numbers (Prime Number):writeCannot be written as the product of two positive integers smaller positive integer except (1), such as 7 or 23.

Mathematical theorems (Mathematical unseen):writeUsing mathematical language to make statements, in particular theoretical system with either correct or incorrect.

Mathematical Proof (Mathematical Proof):writeTo prove the authenticity of mathematical theorems of a series of logical argumentation.Proof must be based on the basic assumption has visited, or other previously have proved the theorem.

Mathematical Conjecture (Mathematical Conjecture):write(also known as the hypothesis) - an is considered reliable mathematical but have yet to prove statements.Produce reliable 'think a guess' 'feeling, is often looked at the special circumstances, are calculated, or the results of mathematical intuition.There are some mathematical conjecture is correct is not recognized by people.

The Twin prime number (Twin Primes):writeA pair of prime Numbers for 2, such as 5, 7 or 41, 43.

Conflict of interest statement

The author statement, the study is in no may be interpreted as potential conflicts of interest under the condition of commercial or financial relationships.

footnotes

1.writeCould be divided into 360 - degree circle approach first appeared in Greece and Egypt astronomers writing, but this is based on 1 hour earlier when the babylonians will be divided into the practice of 60 minutes.There is no doubt that the solar year for 365 days (on average) which is related to the fact that, but please note 5 and 365 = 5 x 73 and 73 are prime, so 365 factoring is much less than 360.

2.writeCorrect mathematics text reading is a kind of ' ' 'active reading, the reader will check the content and calculation example.But if you want to skip the task, can do that, we'll discuss it later.

3.writeTwin prime conjecture due to Leonard Zhang Hemei research in recent years has experienced an alarming breakthrough, but it is still up in the air.About the goldbach conjecture, hull, Mrs. In 2014 proved that every odd number greater than 5 is the sum of the three prime Numbers.

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reference

[1] Du Sautoy, m. 2003. The Music of The Primes. HarperCollins.

[2] Doxiadis, a. 1992. Uncle Petros and Goldbach's Conjecture, Bloomsbury.

[3] Pomerance, C. 2004. "' Prime Numbers and the search for extraterrestrial intelligence, 'in Mathematical Adventures for Students and Amateurs, eds d. Hayes and t. Shubin (M.A.A), 1-4.

[4] Singh, s. 1999. The Code Book. London, The Fourth Estate.