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Number Theory: A Lively Introduction with Proofs, Applications, and Stories
Number theory is one of the oldest and most fascinating branches of mathematics. It deals with the properties and patterns of whole numbers, such as 1, 2, 3, 4, 5, ... Number theory has many applications in science, technology, art, and culture. It also has many unsolved mysteries and intriguing stories that captivate the imagination of mathematicians and non-mathematicians alike.
Number Theory A Lively Introduction With Proofs Applications And Stories PDFpdf
In this article, we will explore some of the main topics and themes of number theory. We will learn about its history, its basic concepts, its famous theorems and conjectures, its applications, and its stories and anecdotes. We will also provide some proofs, examples, and exercises to illustrate the beauty and elegance of number theory.
History of number theory
Number theory has a long and rich history that spans across many civilizations and cultures. The earliest traces of number theory can be found in ancient Egypt, Babylon, China, India, and Greece. These civilizations developed methods for counting, writing, calculating, and solving problems with numbers. They also discovered some remarkable properties and patterns of numbers, such as prime numbers, perfect numbers, Pythagorean triples, Fibonacci numbers, etc.
In the classical era, Greek mathematicians such as Euclid, Eratosthenes, Diophantus, and Archimedes made significant contributions to number theory. Euclid wrote the famous book Elements, which contains many results on geometry and number theory. Eratosthenes invented a method for finding prime numbers called the sieve of Eratosthenes. Diophantus wrote a book called Arithmetica, which contains many problems on finding integer solutions to equations. Archimedes discovered a formula for calculating the approximate value of pi.
In the medieval period, number theory was studied by mathematicians from various regions and religions, such as China, India, Persia, Arabia, and Europe. Some notable names include Brahmagupta, Al-Khwarizmi, Fibonacci, and Fermat. Brahmagupta introduced the concept of zero and negative numbers. Al-Khwarizmi wrote a book on algebra that gave rise to the word algorithm. Fibonacci popularized the Hindu-Arabic numeral system in Europe and introduced the famous sequence that bears his name. Fermat stated many conjectures and results in number theory, such as Fermat's little theorem, Fermat's last theorem, and Fermat's two-square theorem.
In the modern era, number theory witnessed a rapid development and diversification due to the work of many brilliant mathematicians, such as Euler, Gauss, Legendre, Riemann, Dirichlet, Dedekind, Hardy, Ramanujan, Erdős, and Wiles. Euler proved many formulas and identities involving numbers, such as Euler's identity, Euler's totient function, and Euler's product formula for the Riemann zeta function. Gauss wrote a landmark book called Disquisitiones Arithmeticae, which laid the foundations of modern number theory. Legendre introduced the symbol that bears his name and conjectured the law of quadratic reciprocity. Riemann formulated the famous hypothesis that relates the distribution of prime numbers to the zeros of the zeta function. Dirichlet proved a theorem that states that any arithmetic progression with a common difference and a first term that are coprime contains infinitely many prime numbers. Dedekind defined the concept of an ideal and a ring in abstract algebra. Hardy and Ramanujan collaborated on many topics in number theory, such as partitions, modular forms, and asymptotic formulas. Erdős was one of the most prolific mathematicians of all time, who wrote hundreds of papers on various aspects of number theory, such as primes, combinatorics, diophantine equations, and probabilistic methods. Wiles proved Fermat's last theorem after more than three centuries of efforts by many mathematicians.
Basic concepts of number theory
Number theory is based on some fundamental concepts and notions that are essential for understanding and appreciating its results and applications. Here are some of the most important ones:
Divisibility
One of the simplest and most useful concepts in number theory is divisibility. We say that a number a divides another number b, or a is a divisor of b, or b is a multiple of a, if there exists an integer k such that b = ka. We write this as a b. For example, 3 divides 12, or 3 12, because 12 = 3 4. On the other hand, 5 does not divide 12, or 5 12, because there is no integer k such that 12 = 5k.
Divisibility has many properties and consequences that are useful for proving and solving problems in number theory. For example, here are some basic facts about divisibility:
If a b and b c, then a c. This is called transitivity of divisibility.
If a b and a c, then a (b + c) and a (b - c). This is called linearity of divisibility.
If a b and b 0, then a b. This means that a divisor cannot be larger than its multiple in absolute value.
If a = b 0, then a = b. This means that there are only two possible divisors for a nonzero number: itself and its negative.
If a = 1, then a b for any b. This means that 1 and -1 are divisors of every number.
If a b and b a, then a = b 0. This means that two nonzero numbers are equal in absolute value if and only if they divide each other.
If a b and c 0, then (ac) (bc). This means that we can multiply both sides of a divisibility relation by a nonzero factor.
If (ac) (bc) and c 0, then a b. This means that we can cancel a common nonzero factor from both sides of a divisibility relation.
If a 0, then there exist unique integers q and r such that b = qa + r and 0 r < a. This is called the division algorithm, which allows us to write any number as a quotient and a remainder when divided by another nonzero number.
Prime numbers
One of the most important and fascinating concepts in number theory is prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 2, 3, 5, 7, 11, 13, etc. are prime numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 4, 6, 8, 9, 10, etc. are composite numbers.
Prime numbers have many properties and applications that make them very useful and interesting in mathematics and beyond. For example, here are some facts about prime numbers:
There are infinitely many prime numbers. This was proved by Euclid using a clever argument based on contradiction.
Every natural number greater than 1 can be written as a product of prime numbers in a unique way (up to the order of the factors). This is called the fundamental theorem of arithmetic.
The distribution of prime numbers among the natural numbers is not regular or predictable. However, there are some formulas and conjectures that describe how often prime numbers occur on average. For example, the prime number theorem states that the number of primes less than or equal to n is approximately n/logn, where logn is the natural logarithm of n.
There are some special classes of prime numbers that have additional properties or patterns. For example, Mersenne primes are primes of the form 2 - 1, where p is also a prime. The largest known prime number as of June 2021 is a Mersenne prime with 28,862,048 digits.
Prime numbers have many applications in cryptography, which is the science of secure communication. For example, the RSA algorithm, which is widely used for encryption and digital signatures, relies on the fact that it is easy to multiply two large prime numbers, but very hard to factor their product.
Famous theorems and conjectures in number theory
Number theory is full of amazing and surprising results that reveal the hidden beauty and complexity of numbers. Some of these results have been proved by rigorous and elegant arguments, while others remain unproved but highly plausible conjectures. Here are some examples of famous theorems and conjectures in number theory:
Fermat's last theorem
This is one of the most famous and celebrated results in mathematics. It was stated by Pierre de Fermat in the margin of a book in 1637, but he did not provide a proof. He claimed that he had a "marvellous proof" that was too large to fit in the margin. The theorem states that there are no positive integer solutions to the equation x + y = z, where n is any integer greater than 2. For example, there are no positive integers x, y, and z such that x + y = z.
The theorem remained unproved for more than three centuries, despite many attempts by many mathematicians. It was finally proved by Andrew Wiles in 1995, using advanced techniques from algebraic geometry and modular forms. His proof was hailed as a monumental achievement and earned him many awards and honors.
Goldbach's conjecture
This is one of the oldest and simplest unsolved problems in mathematics. It was proposed by Christian Goldbach in a letter to Leonhard Euler in 1742. He conjectured that every even integer greater than 2 can be written as the sum of two prime numbers. For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, etc.
The conjecture has been verified for all even integers up to 4 10, but no general proof has been found. It is widely believed to be true, but no one knows how to prove it. It is one of the Clay Mathematics Institute's Millennium Prize Problems, which offer a $1 million reward for a correct proof or disproof.
Riemann hypothesis
This is another one of the Clay Mathematics Institute's Millennium Prize Problems, and one of the most important and difficult problems in mathematics. It was proposed by Bernhard Riemann in 1859, in a paper that introduced a complex-valued function called the Riemann zeta function. The zeta function is defined as ζ(s) = 1 + 1/2 + 1/3 + 1/4 + ..., where s is any complex number. The hypothesis states that all the non-trivial zeros of the zeta function have a real part equal to 1/2. A zero of a function is a value of the input that makes the output zero. For example, ζ(0) = 0, so 0 is a trivial zero of the zeta function.
The hypothesis has many profound implications for number theory, especially for the distribution of prime numbers. It has been verified for billions of zeros, but no general proof has been found. Many mathematicians consider it to be the most important open problem in mathematics.
Applications of number theory
Number theory is not only a pure and abstract branch of mathematics, but also a very practical and useful one. It has many applications in various fields of science, technology, art, and culture. Here are some examples of applications of number theory:
Cryptography
Cryptography is the science of secure communication, which involves encrypting and decrypting messages using mathematical techniques. Number theory plays a crucial role in cryptography, as it provides methods for generating and manipulating keys, codes, ciphers, and signatures. For example, the RSA algorithm, which is widely used for encryption and digital signatures, relies on the fact that it is easy to multiply two large prime numbers, but very hard to factor their product. Another example is the Diffie-Hellman key exchange, which allows two parties to establish a secret key over an insecure channel, using modular arithmetic and discrete logarithms.
Coding theory
Coding theory is the study of methods for transmitting and storing information efficiently and reliably, using codes that can detect and correct errors. Number theory provides tools for designing and analyzing such codes, such as linear codes, cyclic codes, Reed-Solomon codes, etc. For example, linear codes are based on vector spaces and matrices over finite fields, which are algebraic structures studied in number theory. Reed-Solomon codes are based on polynomials over finite fields, which can correct errors in data transmission and storage.
Combinatorics
congruences, and binomial coefficients. For example, binomial coefficients are numbers that count the ways of choosing k objects from n objects, and they have many properties involving divisibility, congruences, and binomial coefficients.
Stories and anecdotes in number theory
Number theory is not only a rigorous and abstract branch of mathematics, but also a lively and human one. It has many stories and anecdotes that illustrate the history, culture, personality, and humor of number theorists and their discoveries. Here are some examples of stories and anecdotes in number theory:
Math myths
A math myth is a fictional story that introduces an important number theory topic in a friendly, inviting manner. For example, here is a math myth about Fermat's last theorem:
Once upon a time, there was a young prince named Fermat who loved mathematics. He was especially fond of numbers and equations. He spent his days solving puzzles and problems with his tutor, Mersenne.
One day, he came across a book by an ancient mathematician named Diophantus. The book contained many problems about finding integer solutions to equations. Fermat was fascinated by the book and tried to solve all the problems.
He noticed that some of the equations had the form x + y = z, where n was a positive integer. He knew that for n = 2, there were many solutions, such as 3 + 4 = 5. He wondered if there were any solutions for larger values of n.
He tried to find some examples, but he could not find any. He tried to prove that there were none, but he could not find a proof. He became obsessed with the problem and spent many nights thinking about it.
One night, he had a brilliant idea. He found a marvellous proof that there were no solutions for any n > 2. He was so excited that he wrote his proof in the margin of Diophantus's book. However, he made a mistake. He wrote his proof too small and too faintly. He forgot to check his proof carefully and to make a copy of it.
The next day, he was summoned by his father, the king, who had arranged a marriage for him with a princess from another kingdom. Fermat had no choice but to obey his father's wishes and leave his beloved mathematics behind.
He packed his belongings and left for his new home. He forgot to take Diophantus's book with him. The book was left in his old room, where it gathered dust for many years.
Many years later, after Fermat's death, his son found the book and decided to publish it. He noticed his father's note in the margin and was amazed by it. He tried to read his father's proof, but he could not decipher it. He showed it to other mathematicians, but they could not decipher it either.
The note became a legend and a challenge for generations of mathematicians who tried to find Fermat's proof or to find their own proof or counterexample. The problem became known as Fermat's last theorem.
It took more than three centuries for someone to finally prove Fermat's last theorem. His name was Andrew Wiles, and he used advanced techniques from algebraic geometry and modular forms that Fermat could not have known.
No one knows what Fermat's proof was or if he really had one. Some people think that he made a mistake or that he confused the problem with another one. Some people think that he had a valid proof that was lost or destroyed. Some people think that he was joking or teasing his readers.
The mystery of Fermat's last theorem remains one of the greatest stories in mathematics.
Puzzles
A puzzle is a problem or question that challenges one's ingenuity or knowledge. Number theory has many puzzles that are fun and stimulating to solve. For example, here is a puzzle about prime numbers:
Can you find a prime number that is the sum of 17 consecutive prime numbers?
The answer is yes. One such prime number is 953, which is the sum of the 17 consecutive prime numbers from 5 to 83:
5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 83 = 953
This puzzle can be extended to other questions, such as: How many such prime numbers are there? What is the largest or smallest such prime number? What if we change the number of consecutive prime numbers or the starting prime number?
Paradoxes
A paradox is a statement or situation that seems to contradict itself or to defy common sense. Number theory has many paradoxes that challenge one's intuition and logic. For example, here is a paradox about infinity:
Are there more natural numbers or more even numbers?
The answer is that they have the same size, even though it seems that there should be twice as many natural numbers as even numbers. This is because we can pair every natural number with an even number in a one-to-one correspondence, such as:
1 2, 2 4, 3 6, 4 8, ...
This shows that the two sets have the same cardinality, which is a measure of the size of infinite sets. This paradox can be extended to other questions, such as: Are there more rational numbers or more irrational numbers? Are there more real numbers or more complex numbers? Are there more algebraic numbers or more transcendental numbers?
Personalities
A personality is a person's characteristic pattern of thinking, feeling, and behaving. Number theory has many personalities that reflect the diversity and creativity of number theorists and their discoveries. For example, here is a personality of Paul Erdős:
analysis. He had no permanent home, no job, no family, and no belongings. He traveled around the world, staying with his friends and collaborators, and working on mathematical problems. He was generous, eccentric, and passionate about mathematics. He had his own language and terminology, such as calling children "epsilons", God "the Supreme Fascist", and death "being on the other side". He had a legendary stamina and could work for hours without sleep or food. He was addicted to caffeine and amphetamines, which he claimed helped him to think better. He collaborated with more than 500 mathematicians, and wrote more than 1500 papers on various topics in mathematics. He had a large influence on the development of mathematics in the 20th century.
Conclusion
Number theory is a fascinating and rich branch of mathematics that explores the properties and patterns of whole numbers. It has a long and colorful history that spans across many civilizations and cultures. It has many basic concepts that are essential for understanding and appreciating its results and