What is Number Theory in math?
Number theory is the study of the properties of the positive integers or whole numbers and is one of the oldest areas of mathematics. The most basic areas are what numbers evenly divide what other numbers, which leads to the prime numbers. Number theory is also one of the most difficult areas of mathematics, it has been said many times that if a problem remains unsolved for hundreds of years, it is almost always in number theory.
What are number theory concepts?
Number theory studies integers, that is, whole numbers, and their relationships. Principle concepts include square numbers and higher powers of numbers, prime numbers, divisibility of numbers, and greatest common divisors. Many of the questions in number theory have to do with equations.As equations involving integers can be interconverted with equations involving rational numbers (quotients of integers), number theory deals with rational numbers as well.Some of the questions in number theory derive from geometry. An example of that is the classification of Pythagorean triples, integers a, b, and c such that [math]a^2+b^2=c^2.[/math] They describe right triangles with integral sides. Perhaps this classification was the first question in number theory, apparently answered over a thousand years before Pythagoras.One of the earliest tools for studying number theory is what is called the Euclidean algorithm. It's not due to Euclid, but he did mention it in his Elements. It was also known in ancient China and India, and was used in the Chinese Remainder Theorem (which was also known in India but not to the ancient Greeks). It's closely related to continued fractions.Looking for solutions in integers for equations isn't restricted to single linear equations (which requires the Euclidean algorithm), but also to quadratic equations, higher degree equations, and simultaneous equations. Many people throughout the centuries have worked on their solutions. Euler, in the 18th century, developed the concept of congruence of numbers modulo n, and that concept is now used throughout number theory. The quadratic equations are very interesting. See especially Pell's equation, quadratic reciprocity, and quadratic forms.Analysis, that is the part of mathematics relating to concepts of differentiation, integration, and infinite series, can be used to answer questions in number theory. The branch of number theory that uses analysis is called analytic number theory. That's advanced number theory a lot in the last two hundred years.Algebraic number theory uses concepts of modern algebra such as number fields, groups, rings, and Galois theory.One might thing that with such an old subject in mathematics with a rich history that covers four thousand years that there's nothing left to do. That's not the case. It was one of the most active mathematical fields in the 20th century, and it appears that it will be one again in the 21st century.
What are the basic components of number theory?
André Weil, one of the leading mathematicians of the 20th century, wrote a book he called “Basic Number Theory”.Seeing the title, one expects the book to begin with the usual elements – arithmetic operations, prime numbers, unique factorization and so on.No such luck. Part I (“Elementary Theory”) starts with a chapter on locally compact fields, and proceeds to discuss adeles, zeta functions and the Riemann-Roch theorem. To be clear, none of that is undergrad material, and most graduate students in fields that aren't number theory may not even know what adeles are.But such is the nature of this domain. The book’s title is probably somewhat tongue-in-cheek, but not entirely: those really are the basics of the modern theory of numbers. Researchers in Number Theory have to master class field theory, for example, which was developed around the 1900’s. It's “old”, but decidedly not “easy”. Weil’s book is essentially an introduction to this theory.Many problems in number theory require no more than addition and multiplication to describe. The solutions to those problems, however, often require huge parts of the most modern machinery of mathematics.So number theorists “start with” some very elementary definitions, as you said, but they rarely do anything with those definitions alone. Within two hours of reading about them, they move on to more advanced perspectives.If you wish to study number theory seriously, you have a lifetime of studying ahead of you, but it's a truly wonderful journey.
Number Theory Questions?
(a) Multiply both sides of the equation by a^(φ(n) - 1): ax = b (mod n) ==> a^(φ(n)) x = a^(φ(n) - 1) b (mod n). However, Euler's Theorem ==> a^(φ(n)) = 1 (mod n), because gcd(a, n) = 1. Therefore, x = a^(φ(n) - 1) b (mod n). (b) Using part a: (i) x = 3^(φ(26) - 1) * 5 = 3^(12 - 1) * 5 = (3^3)^2 * 3^2 * 5 = 1 * 45 = 19 (mod 26). The others are done similarly. I hope this helps!
Number theory questions?
1. This is correctly answered already by Northstar and JB above. 2. If n > 4 is composite, since GCD(n, n-1) = 1, then either 2a) n = p² (a square of a prime), 2 < p < n-1 (4 = 2²) or 2b) n = r*s, r and s are distinct integers where 1 < r < s < n-1 In case 2a) both p and 2p are to be found among the factors of (n-1)! since 2 < p implies 2p ≤ n-1 < n = p² In case 2b) similarly both r and s participate among the factors of (n-1)!. Either way n divides (n-1)! 3. p² - q² = (p - q)(p + q), here both factors are even since p ≥ q ≥ 5 and their difference (p + q) - (p - q) = 2q is even also. Then at least one of the factors is divisible also by 4 since p² - q² = (p - q)(p + q) = (p - q)((p - q) + 2q) and assuming the opposite the difference 2q would be a multiple of 4 - impossible for prime q. Hence the product is divisible by 8. Since both p and q are greater than 3 we have p ≡ ±1 (mod 3) and q ≡ ±1 (mod 3) Whatever sign combination we take above at least one of p + q ≡ 0 (mod 3) or p - q ≡ 0 (mod 3), hence the product of both is divisible by 3 also. 4. Every prime p, except 2 and 3 is congruent to 1 or 5 by modulo 6: p ≡ ±1 (mod 6) implies p² ≡ 1 (mod 6), p² + 8 ≡ 9 ≡ 3 (mod 6), so p² + 8 is prime only for p = 3. But then p³ + 4 = 31 - also prime.
What are some good contest math books for basic number theory?
Methods of Solving Number Theory Problems [Ellina Grigorieva](2018)250 Problems in Elementary Number Theory [Waclaw Sierpinski](1970)1001 Problems in Classical Number Theory [Jean Marie De Koninck, Armel Mercier](2007)
Can you suggest a good book to study basic number theory?
This is more of an introductory text. Good to read when transitioning from high school to college. It builds good intuition and basic proof techniques.Amazon.com: Elementary Number Theory (9780073383149): David Burton: BooksGood text book for collegiate level material. Really wonderful. Haven't read this completely though, since I am not a math major.Amazon.com: An Introduction to the Theory of Numbers (9780471625469): Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery: BooksOf course, there are more classics which you can find mentioned here:Best book ever on Number Theory
Elementary number theory?
Since 3, 7, and 8 are relatively prime, it will do to show that each of these numbers separately divides a^6 – 1. For 3: gcd(a, 42) = 1 ∴ gcd(a, 3) = 1 Fermat's Little Theorem says that if p is prime and gcd(a, p) = 1, then a^(p – 1) ≡ 1 (mod p). Applying this with p = 3, we get a^2 ≡ 1 (mod 3). ∴ a^6 ≡ 1 (mod 3) ∴ 3 divides a^6 – 1. For 7: gcd(a, 42) = 1 ∴ gcd(a, 7) = 1 ∴ a^6 ≡ 1 (mod 7), using Fermat's Little Theorem again ∴ 7 divides a^6 – 1. For 8: gcd(a, 42) = 1 ∴ a is odd, so it is congruent to 1, 3, 5, or 7 modulo 8. Now, observe that as 1^2 = 1, 3^2 = 9, 5^2 = 25, and 7^2 = 49 are all congruent to 1 modulo 8, a^2 ≡ 1 (mod 8.) ∴ a^6 ≡ 1 (mod 8) ∴ 8 divides a^6 - 1. This completes the proof.
How can I get started to learn number theory?
Let’s get started with number theory. By numbers, I mean positive numbers for simplicity.Get started with the definition of divisibility that if a | b (read a divides b) then b = ak for some integer k.Now you might think of “not divides” too. That is remainder. So far it is elementary math you learned in school.Learn the concept of factor and multiple of a number. If a is a factor of b, then a divides b and b is a multiple of a. Simple. Right?Observe that 1 and the number itself are the two factors for any number. So the number of factors for a > 1 is at least 2.Now consider the numbers a > 1 which have exactly two factors. They are called prime numbers. Others are composite. 1 is neither prime nor composite. That’s it. Is the number of primes infinite? Answer it!Can you write any number as a product of one or more prime factors? Is it unique? That is prime factorization and the fundamental theorem of arithmetic. Prime factorization is hard and you learned one of the important number theoretic problems.How about counting the number of factors of a given number? How about finding the sum of all of them? (Given only their prime factorization).Learn the concept of Greatest Common Divisors and Least Common Multiple. Finding GCD of two given numbers - you’ll learn one of the ancient and efficient algorithms which is used even today - Euclid’s algorithm to find GCD.Given a number n, how about finding the number of numbers 1 <= m < n such that they gcd(m,n) = 1? That is Euler’s phi function. You have already made it this far! Congratulations.Now let’s go to modular arithmetic. a congruent to b modulo m means m divides a-b. See it is just a fancy notation.Then read about Fermat’s little theorem, Euler’s theorem. You’ll also learn about Bezout’s identity, the concept of modular inverses etc.How about solving some equations involving congruences? (involving modular arithmetic). Consider linear ones for simplicity. Is it possible to have a solution all the time? Another interesting problem in Cryptography is Discrete logarithm problem.Great, this is the beginning of the number theory. It is just the tip of the iceberg. There are lot of interesting unsolved problems like Twin Prime Conjecture, Goldbach conjecture etc. Others have suggested various resources. Use them and good luck in your journey in the land of numbers :)