Elementary number theory is a branch of mathematics that studies the laws of numbers, especially the properties of integers.

It is one of the oldest branches of number theory. It takes arithmetic method as its main research method, and its main contents include integer divisibility theory, indefinite equation, congruence formula and so on.

Pythagoras in ancient Greece was the pioneer of elementary number theory. He and his school are devoted to the study of some special integers (such as affinity number, perfect number, polygon number) and special indefinite equations.

In the 4th century BC, Euclid's Elements of Geometry initially established the theory of integer divisibility through 102 propositions. His proof of "there are infinitely many prime numbers" is regarded as a model of mathematical proof.

In the 3rd century AD, Diophantine studied many indefinite equations and designed ingenious solutions respectively, so later people called indefinite equations Diophantine equations. Since17th century, the work of P.de Fermat, L. Euler and C.F. Gauss has greatly enriched and developed the content of elementary number theory.

In ancient China, brilliant achievements were made in the study of elementary number theory, which were recorded in ancient literature, such as Zhou Bian Shu Jing, Sun Zi Shu Jing, Zhang Qiujian Shu Jing, Shu Shu Jiu Zhang and so on. Sun Tzu's theorem predates Europe by 500 years. This theorem is often called China's remainder theorem in the west, and Qin's technique of seeking greatness and broadening is also famous all over the world.

Elementary number theory is not only the basis of learning pure mathematics, but also an important tool for many disciplines. Its applications are various, such as computer science, combinatorial mathematics, cryptography, information theory and so on.

For example, public key system is an important application of number theory in cryptography. Elementary number theory is to study number theory with elementary and simple methods.

In addition, there is analytic number theory. ), algebraic number theory (using algebraic structure to study number theory).

At the beginning, prime number theory used simple reasoning methods to study the properties of integers, and prime numbers were the most fascinating. I don't know how many mathematicians have worked hard for it! Studying the properties of prime numbers is a very important aspect in number theory! The so-called prime number is a positive integer, and there are no other factors except itself and 1.

Prime numbers are like atoms of positive integers. The famous gauss "unique decomposition theorem" says that any integer. It can be written as the product of a series of prime numbers.

But so far, there is still no universal and special formula that can represent all prime numbers. Therefore, two famous conjectures about prime numbers in number theory are very difficult: 1 Goldbach conjecture: the content is that "all even numbers greater than 2 can be expressed as two prime numbers". This problem was solved by the German mathematician C. Goldbach (1690- 1764) in 65438+ years.

On June 30th of the same year, Euler replied that this conjecture may be true, but he could not prove it. Since then, this mathematical problem has attracted the attention of almost all mathematicians.

Goldbach conjecture has therefore become an elusive "pearl" in the crown of mathematics. "In contemporary languages, Goldbach conjecture has two contents, the first part is called odd conjecture, and the second part is called even conjecture.

Odd number conjecture points out that any odd number greater than or equal to 7 is the sum of three prime numbers. Even number conjecture means that even numbers greater than or equal to 4 must be the sum of two prime numbers. "

(Quoted from Goldbach conjecture and Pan Chengdong) Goldbach conjecture seems simple, but it is not easy to prove, which has become a famous problem in mathematics. In 18 and 19 centuries, all number theory experts did not make substantial progress in proving this conjecture until the 20th century.

It is directly proved that Goldbach's conjecture is not valid, and people adopt "circuitous tactics", that is, first consider expressing even numbers as the sum of two numbers, and each number is the product of several prime numbers. If the proposition "every big even number can be expressed as the sum of a number with no more than one prime factor and a number with no more than b prime factors" is recorded as "a+b", then the Coriolis conjecture is to prove that "1+ 1" holds.

1900, Hilbert, the greatest mathematician in the 20th century, listed Goldbach conjecture as one of the 23 mathematical problems at the International Mathematical Congress. Since then, mathematicians in the 20th century have "joined hands" to attack the world's "Goldbach conjecture" fortress, and finally achieved brilliant results.

In the 1920s, people began to approach it. 1920, the Norwegian mathematician Bujue proved by an ancient screening method that every even number greater than 6 can be expressed as (9+ 9).

This method of narrowing the encirclement is very effective, so scientists gradually reduce the number of prime factors of each number from (99) until each number is a prime number, thus proving Goldbach's conjecture. 1920, Bren of Norway proved "9+9".

1924, Rademacher proved "7+7". 1932, Esterman of England proved "6+6".

1937, Ricei of Italy proved "5+7", "4+9", "3+ 15" and "2+366" successively. 1938, Byxwrao of the Soviet Union proved "5+5".

1940, Byxwrao of the Soviet Union proved "4+4". 1948, Hungary's benevolence and righteousness proved "1+c", where c is the number of nature.

1956, Wang Yuan of China proved "3+4". 1957, China and Wang Yuan successively proved "3+3" and "2+3".

1962, Pan Chengdong of China and Barba of the Soviet Union proved "1+5", and Wang Yuan of China proved "1+4". 1965, Byxwrao and vinogradov Jr of the Soviet Union and Bombieri of Italy proved "1+3".

1966, China and Chen Jingrun proved "1+2" [in popular terms, it means even number = prime number+prime number * or even number = prime number+prime number (Note: the prime numbers that make up even numbers cannot be even numbers, but only odd numbers. Because there is only one even prime number in the prime number, that is 2. )

]。 The "s+t" problem refers to the sum of the products of S prime numbers and T prime numbers. The main methods used by mathematicians in the 20th century to study Goldbach's conjecture are screening method, circle method, density method and triangular sum.

2. Some basic knowledge of number theory

If it is limited to elementary number theory, then the research object of elementary number theory is relatively narrow, generally integer or even natural number. Advanced research even scored above this limit.

Elementary number theory studies negative integers in principle, such as Diophantine equation. And if we only talk about the most basic divisibility and prime numbers, it is enough to study natural numbers.

The most basic tool of elementary number theory is divisible congruence, that is, 6 is an integer divisible by 2, that is, 6 can be divisible by 2; 6 divided by 4 is a fraction, which means that 6 is not divisible by 2. Congruence refers to dividing two numbers by the same number (called modulus) to see if the same remainder is obtained. For example, for modules 7, 2 and 9, 3 and 6 are different.

Attached concepts include the greatest common divisor and so on. Euclid algorithm is the basic method to find the greatest common divisor.

The development of higher direction can include primitive roots, quadratic residues, Pell equation, number theory function, prime number distribution, graphic lattice and so on. In short, elementary number theory uses no more tools than elementary analysis.

3. How to learn elementary number theory?

Elementary number theory, also known as integer theory, mainly studies the properties of integers and the integer solutions of equations. It is a very important branch of basic mathematical theory. Because the problems in elementary number theory are easy to understand, it can attract people's attention more than any other branch of mathematics. Many important ideas, concepts, methods and skills in modern mathematics are constantly enriched and developed from the in-depth study of the properties of integers. This course has 3 credits and 54 hours.

This course is divided into five chapters, introducing divisibility theory, indefinite equation, congruence theory and continued fraction respectively, focusing on four modules: integer divisibility, indefinite equation, unary congruence theory and square residue. The main task of this course is to enable students to deepen their understanding of integers and their properties, and on the other hand, to enable students to master the basic research methods and skills of elementary number theory, which is conducive to students' better elementary mathematics teaching.

The text textbook of this course is equipped with a series of examples and exercises according to the difficulty of knowledge points, and 20 hours of IP courseware is compiled for students to learn online, and a series of online tutoring exercises are compiled. Integer divisibility module requires mastering the concepts and properties of integer divisibility, common factor and prime number, skillfully using alternate division to find the greatest common factor and the least common multiple of two integers, and deeply understanding the residue theorem and basic theorem of arithmetic. It can find a simple prime table by filtering. The pigeon hole principle will be used to prove some simple problems about integers being multiples of specific integers. Indefinite equation module requires keeping in mind the condition that binary linear indefinite equation has integer solution and the form of integer solution of binary linear indefinite equation, and mastering the method of finding integer solution of binary linear indefinite equation by residue theorem (division). Knowing the condition that the multivariate linear indefinite equation has a solution, we can find the integer solution of the univariate multivariate (ternary) linear indefinite equation; If we know the form of integer solution of indefinite equation, we can derive integer solution with this form and prove some simple related problems. Master the method of judging congruence, and understand the definition and properties of Euler function; Understand euler theorem and Fermat Theorem, and master the judgment method of cyclic decimal; Master the condition that a congruence has a solution, and master the solution of a congruence skillfully; Master the simple application of China's remainder theorem and the solution method of simple congruence equations; Know how to judge the number of solutions of high-order congruence, how to solve the relationship between high-order congruence, modular integer congruence and modular prime congruence, and how to find simple (3,4) congruence. Square residue module requires understanding the general form of quadratic congruence, the relationship between modular integer congruence and modular prime power congruence, and the concepts of square residue and square no residue. Understand the Euler judgment method of single prime square residue and square non-residue, and understand the number of single prime square residue and square non-residue; Understand the definitions and properties of Legendre symbols and Jacobian symbols, and master the use of Legendre symbols and Jacobian symbols to judge the existence of congruence solutions; Master the conditions for quadratic congruence of non-prime modules to have solutions and related conclusions about the number of solutions; The conditions for indefinite equations to have integer solutions to prime numbers P will be discussed. Master the simple method of finding the original root; Will use the original root to get the method of integer simplified remainder system; Master the application of indicators (discuss the conditions for congruence to have solutions and the number of solutions). In the study of many number theory problems, China is in a leading position. Famous mathematicians of the older generation, such as Hua, Ke Zhao and Min Sihe, have made brilliant achievements, especially Professor Hua's achievements in analytic number theory are universally recognized. After the 1960s, famous mathematicians such as Chen Jingrun, Wang Yuan and Pan Chengdong also made international leading achievements on Goldbach conjecture and other issues. Our only advice is to do and practice. Learning elementary number theory is like learning a new course of practice and practical technology. Practice more, preferably one lesson at a time, or even a certain theory (or example). If you don't know anything, you can read books or examples or do supporting exercises repeatedly, and maybe you will be suddenly enlightened. Learning this course well does not require a high professional knowledge background. As long as you can take the time to study hard, some formulas need to be memorized and used flexibly to grasp the key points.

Learning a course requires certain skills, learning to classify and summarize, and grasping joint points. Unconsciously aroused your interest and enthusiasm in learning and exploring this course, and it will be more handy to learn.

4. The content of elementary number theory.

Elementary number theory has the following parts:

1. divisibility theory. This paper introduces the basic concepts such as divisibility, factor, multiple, prime number and composite number. The main achievements of this theory are: unique decomposition theorem, Pei tree theorem, Euclid's transition division, fundamental theorem of arithmetic, proof of infinite prime number, etc.

2. Congruence theory. Mainly from Gaussian arithmetic research. The concepts of congruence, primitive root, exponent, square residue and congruence equation are defined. Main achievements: Quadratic Reciprocity Law, euler theorem, Fermat's Last Theorem, Wilson's Theorem, Sun Tzu's Theorem (China's Remainder Theorem), etc.

3. Continued fraction theory. The concept and algorithm of continued fraction are introduced. In particular, the continued fraction expansion of integer square root is studied. Main achievements: cyclic continued fraction expansion, optimal approximation problem, solving Pell equation.

4. Indefinite equation. This paper mainly studies the indefinite equations corresponding to the lower algebraic curve, such as the quotient height theorem of Pythagorean equation and the continued fraction solution of Pell equation. It also includes the solution of the fourth Fermat equation and so on.

5. Number theory function. Such as Euler function, Mobius transformation and so on.

6. Gaussian function. The first level is called mathematical concept, which is a form of thinking that reflects the essential attributes of objects. In the process of cognition, human beings rise from perceptual knowledge to rational knowledge, and abstract and summarize the essential characteristics of perceptual things, which becomes a concept. The language form of expressing concepts is words or phrases. Scientific concepts, especially mathematical concepts, are more stringent, and at least three conditions must be met: specificity, accuracy and verifiability. For example, "twin prime numbers" is a mathematical concept.

The second level is called mathematical proposition, which is a sentence to judge the relationship between a series of mathematical concepts. A proposition is either true or false (this is guaranteed by law of excluded middle in logic). A true proposition contains theorems, lemmas, inferences, facts, etc. A proposition can be an existential proposition (expressed as "being"). . . "), or it can be a full-name proposition (expressed as" for everything ". ..")。 The third level is called mathematical theory, which combines methods, formulas, axioms, theorems and principles into a system. For example, "elementary number theory" consists of axioms (such as equality axioms), theorems (such as Fermat's last theorem), principles (such as one-to-one correspondence principle of pigeon hole principle) and formulas. In mathematical proof, the full name proposition can not always be judged by enumeration, because mathematics sometimes faces infinite objects, and it can never enumerate every situation one by one. Incomplete induction is not feasible in mathematics, which only recognizes deductive logic (mathematical induction, transfinite induction, etc.) ).

5. Find the basic concepts, theories and theorems of elementary number theory. The more complete, the better.

The first chapter is about the algorithm of number theory: 1. 1 maximum common divisor and minimum common multiple 1.2 prime number: 1.3 integer solution of equation ax+by=c and its application 1.4 to find the maximum common divisor and minimum common multiple/kloc. Beginif b=0 then gcdd:=aelse gcd:=gcd(b, a mod b); End; 2. Algorithm 2: the least common multiple acm=a*b div gcd(a, b); 3. Algorithm 3: Extended Euclidean algorithm, find gcd(a, b) and integer x, y function exgcd(a, b: longint; var x，y:longint):longint； vart:longint； Beginif b=0, then begin result:= a；; x:= 1； y:= 0； endelsebeginresult:=exgcd(b，a mod b，x，y)； t:= x； x:= y； y:= t-(a div b)* y； End; End; (theoretical basis: gcd (a, b) = ax+by = bx1+(a mod b) y1= bx1+(a-(a div b) * b) y1= ay/kloc-. const maxn = 1000； varpnum，n:longint； P: array [1... maxn] of longint; function is prime(x:longint):boolean； Var i: integer; I: = 1 to pnum doifsqr (p [I]) 0) at first. Now cylinder C is full of water. Ask if it is possible to measure D liters of water in cylinder C (c>d>0). If so, please list a plan. Algorithm analysis: the process of measuring water is actually to pour back and forth, and there are always the following points when pouring: 1. There is always a water in a tank that hasn't changed; 2. One cylinder is filled or the other cylinder is emptied; 3. Cylinder C only plays a transit role, and its own volume is not limited except that it must be enough to hold all the water of Jane A and Jane B. The procedure is as follows: program MW; Typenode=array[0..longint's1]; Vara, b, c: nodes; d，step，x，y:longint； Function exgcd(a, b: longint; var x，y:longint):longint； var t:longint； beginif b = 0 thenbegncd:= a； ; x:= 1； y:= 0； endelsebegnxcd:= ex gcd(b，a mod b，x，y)； t:= x； x:= y； y:= t-(a div b)* yend； End; Process equation (a, b, c: longint; var x0，y0:longint)； var d，x，y:longint； begind:=exgcd(a，b，x，y)； If c mod d>0 thenbeginwriteln ("no answer"); Stop; end elsebeginx 0:= x *(c div d)； y0:= y *(c div d)； End; End; Process filling (variables a, b: nodes); var t:longint； Begin if a [1] 0 then repeat if a [1] = 0 and then fill(c, a) elseif b[ 1]=b[0] and then fill(b, c) else fill(a, b); Inc (step); Writeln (steps: 5,':', a[ 1]:5, b[ 1]:5, c [1]: 5); Until c [1] = delserepeatif b [1] = 0 and then fill (c, b) elseif a[ 1]=a[0] and then fill (a, c) otherwise fill (b, a); Inc (step); Writeln (steps: 5,':', a[ 1]:5, b[ 1]:5, c [1]: 5); Until c [1] = d; End. 1.4 find a b mod n 1. Algorithm 8: Find a b mod n function f (a, b, n: longint): longint by direct iteration; var d，I:longint； begin d:= a； For i:=2 to b do d:= d mod n * a；; d:= d mod n； f:= d； End; 2. Algorithm 9: Accelerated iteration function f (a, b, n: longint): longint; var d，t:longint； begin d:= 1； t:= a； And b>0 do begin if t= 1 then begin f:= d；; Exit end; If b mod 2 = 1, then d: = d * t mod n; b:= b div 2； t:= t * t mod n； End; F:=d end; Exercise: 1 Memorize the above algorithm, memorize it.

6. Little knowledge of mathematics

1. In life, we often use the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Do you know who invented these numbers? These digital symbols were first invented by ancient Indians, and then spread to * * *, and then from * * * to Europe. Europeans mistakenly think that it was invented by * * * people, so it is called "* * * number". Because it has been circulating for many years, people still call them * * *. Now, the number * * * has become a universal digital symbol all over the world.

2. Nine Jiu Ge is the multiplication formula we use now. As early as the Spring and Autumn Period and the Warring States Period BC, Jiujiu songs have been widely used by people.

In many works at that time, there were records about Jiujiu songs. The original 99 songs started from "99 8 1" to "22 gets 4", with 36 sentences.

Because it started with "998 1", it was named 99 Song. The expansion of Jiujiu songs to "11" was between the 5th century and10th century.

It was in the 13 and 14 centuries that the order of Jiujiu songs changed from "one to one" to "9981". At present, there are two kinds of multiplication formulas used in China. One is a 45-sentence formula, usually called "Xiao Jiujiu"; There is also a sentence 8 1, which is usually called "Big Uncle Nine".

3. The circle is a seemingly simple but actually wonderful circle. The ancients first got the concept of circle from the sun and the moon on the fifteenth day of the lunar calendar.

Even now, the sun and the moon are used to describe some round things, such as the Moon Gate, Qin Yue, Moon Shell, Sun Coral and so on. Who drew the first circle? The stone balls made by the ancients more than 100 thousand years ago are quite round.

As mentioned earlier, Neanderthals 18000 years ago used to drill holes in animal teeth, gravel and stone beads, some of which were very round. Neanderthals drilled holes with pointed devices, but they couldn't get in on one side and then drilled from the other.

The tip of the stone tool is the center of the circle, and half of its width is the radius. Turn around and you can drill a round hole. Later, in the pottery age, many pottery were round.

Round pottery is made by putting clay on a turntable. When people start spinning, they make round stones or ceramic cocoons.

Banpo people (in Xi 'an) built round houses 6000 years ago, with an area exceeding 10 square meter. The ancients also found that rolling logs was more economical.

Later, when they were carrying heavy objects, they put some logs under big trees and stones and rolled them around, which was of course much more labor-saving than carrying them. Of course, because the log is not fixed under the weight, you have to roll the log rolled out from the back to the front and pad it under the front of the weight.

About 6000 years ago, Mesopotamia made the world's first wheel-a round board. About 4000 years ago, people fixed round boards under wooden frames, which was the original car.

Because the center of the wheel is fixed on a shaft, and the center of the wheel is always equal to the circumference, as long as the road surface is flat, the car can move forward in a balanced way. You can make a circle, but you don't necessarily know its nature.

The ancient Egyptians believed that the circle was a sacred figure given by God. It was not until more than 2,000 years ago that China's Mozi (about 468- 376 BC) defined the circle: "One China has the same length".

It means that a circle has a center and the length from the center to the circumference is equal. This definition is 100 years earlier than that of the Greek mathematician Euclid (about 330 BC-275 BC).

Pi, the ratio of circumference to diameter, is a very strange number. The Book of Weekly Calculations says that "the diameter is three times a week", and pi is considered to be 3, which is only an approximate value.

When the Mesopotamians made the first wheel, they only knew that pi was 3. In 263 AD, Liu Hui of Wei and Jin Dynasties annotated Nine Chapters of Arithmetic.

He found that "the diameter is three times that of a week" is just the ratio of the circumference to the diameter of a regular hexagon inscribed in a circle. He founded secant technology, and thought that when the number of inscribed sides of a circle increased infinitely, the circumference was closer to the circumference of a circle.

He calculated the pi = 3927/1250 of the inscribed circle of the regular 3072-sided polygon. Would you please convert it into decimal and see what it is? Liu Hui applied the concept of limit to solving practical mathematical problems, which is also a great achievement in the history of mathematics in the world. Zu Chongzhi (AD 429-500) continued to calculate on the basis of previous calculations, and found that the pi between 3. 14 15926 and 3. 14 15927 was the earliest value in the world accurate to seven decimal places. He also used two fractional values to express pi: 22/7 is called approximate ratio.

Please change these two fractions into decimals and see how many decimals are the same as the known pi today. In Europe, it was not until 1000 years later16th century that the Germans Otto (A.D. 1573) and Antoine Z got this value. Now that there is an electronic computer, pi has been calculated to more than 10 million after the decimal point.

4. Besides counting numbers, mathematics needs a set of mathematical symbols to express the relationship between numbers and shapes. The invention and use of mathematical symbols are later than numbers, but they are much more numerous.

Now there are more than 200 kinds in common use, and there are more than 20 kinds in junior high school math books. They all had an interesting experience.

For example, there used to be several kinds of plus signs, but now the "+"sign is widely used. +comes from the Latin "et" (meaning "and").

/kloc-in the 6th century, the Italian scientist Nicolo Tartaglia used the initial letter of "più" (meaning "add") to indicate adding, and the grass was "μ" and finally became "+". The number "-"evolved from the Latin word "minus" (meaning "minus"), abbreviated as m, and then omitted the letter, it became "-".

It is also said that wine merchants use "-"to indicate how much a barrel of wine costs. After the new wine is poured into the vat, a vertical line is added to the "-",which means that the original line is erased, thus becoming a "+"sign.

/kloc-In the 5th century, German mathematician Wei Demei officially determined that "+"was used as a plus sign and "-"was used as a minus sign. Multipliers have been used for more than a dozen times, and now they are commonly used in two ways.

One is "*", which was first proposed by the British mathematician Authaute at 163 1; One is "",which was first created by British mathematician heriott. German mathematician Leibniz thinks: "*".