What is Number theory: Definition and 471 Discussions

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).
Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation).
The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is commonly preferred as an adjective to number-theoretic.

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  1. R

    Elementary number theory - proving primality

    Homework Statement if an integer n >= 2 and if n divides ((n-1)! +1) prove that n is prime. Homework Equations a divides b iff b = ma for integers a, b, m. The Attempt at a Solution by contrapositive: Assume n is not prime. Then we have by definition of divisibility...
  2. A

    Does one need to know elementary number theory to study Abstract Algebra?

    It's been some time that I've been studying abstract algebra and now I'm trying to solve baby Herstein's problems, the thing I have noticed is that many of the exercises are related to number theory in someway and solving them needs a previous knowledge or a background of elementary number...
  3. A

    Proving the Existence of a Subset with Divisible Sum in Number Theory

    Homework Statement Suppose n is a natural number and A is a subset of natural number with n elements. Prove that a subset of A exists that the sum of its elements is dividable by n. The Attempt at a Solution well, This problem is harder than I can solve it. I first tried to use the...
  4. R

    [Number Theory] Prove (x^2 - y^2) is not equal to 6.

    Homework Statement Prove the following proposition: For any positive integers x and y, (x^2 - y^2) is not equal to 6.Homework EquationsThe Attempt at a Solution I'll try to prove using contradition. Assume x^2 - y^2 = 6. (x+y)(x-y) = 6 (x+y)=6 and (x-y)=1 (OR) (x+y)=1 and (x-y)=6 (OR)...
  5. T

    Elementary Number Theory Proof

    Homework Statement If 3 | m^2 for some integer m, then 3 | m. Homework Equations a | b means there exists an integer c such that b = ca. The Attempt at a Solution I realize that this is a corollary to Euclid's first theorem, and that there are plenty of ways to solve this. However, I...
  6. V

    Looking for a good number theory book(

    Here is my situation, In short I want a number theory book that doesn't assume knowledge of previous number theory but assumes all knowledge of mathematics. I have been knowing multivariable calculus since about 4 years(learned it from Thomas 4th edition(this book was not like the later...
  7. H

    Some question about number theory

    How to prove that if a-c | ab+cd then a-c | ad+cb is correct?? And how to prove the gcd(a^2+b^2, a+b) is 1 or 2. where gcd(a,b)=1.
  8. P

    How Does the Möbius Function Influence Summations in Number Theory?

    First note that the operation * denotes the Dirichlet product, and µ denotes the Möbius function. Ok so here is the problem: Let f(x) be defined for all rational x in 0≤x≤1 and let F(n)=\sum_{k=1}^nf(\frac{k}{n})\;\;\;\;\; and \;\;\;\;\;\; G(n)=\sum_{k=1,\;(k,n)=1}^nf(\frac{k}{n}). Prove that...
  9. P

    What puts the analysis in analytic number theory?

    I'm interested in analytic number theory and from what little I understand of it complex analysis will be more important than real analysis(measure theory). Thus I will be taking a year of graduate complex analysis this fall, however, I do also have the option of taking a year of graduate real...
  10. BloodyFrozen

    Number Theory, Linear & Abstract Algebra

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  11. A

    Number theory proof: Unique determination of a recursively defined function

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  12. L

    Would Number Theory Collapse if Riemann's Hypothesis is Proven True?

    If Riemann's Hypothesis is proved as true, would number theory collapse?
  13. Q

    One problem about number theory

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  14. R

    Good self-teaching book for elementary and advanced number theory?

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  15. T

    NUMBER THEORY: Show that 8^900 - 7 is divisable by 29 Help

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  16. M

    Number theory: Find the last three digits

    Prove that the last three digits of n^100 can be only: 000, 001, 376, or 625. I can easily show that the last digit is either 0, 1, 6 or 5 because n^100=((n^25)^2)^2, so if our last three digits are 100a+10b+c, with a, b, c belonging to {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, any digit for c...
  17. M

    Number Theory: Division with remainder of factorials

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  18. P

    Curious Number Theory Roadblock

    So basically I am trying to prove that the sum ∑1/2k from k=1 to n is a fraction of the form odd/even, that is to say that the denominator will contain more 2's than the numerator. Now I'm almost positive this is true, and I suppose it might be more tractable to consider the stronger...
  19. srfriggen

    Courses Number Theory. What can I expect from such a course

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  20. M

    Number Theory - Find Remainder when dividing by 17

    Number Theory -- Find Remainder .. when dividing by 17 Homework Statement Find the remainder when 3^24*5^13 is divided by 17. Homework Equations I know that 3^24 = 16 (mod 17) and calculated that 5^13 mod 17 = 3 (mod 17) The Attempt at a Solution BUT, I'm completely unsure...
  21. M

    Number theory question - binary trees

    Here's the question. Starting with an integer a≥2, we write on its left, below it, the number a+1, and on its right, below it, the number a^2, and obtain four numbers, to which we continue the process. We thus obtain a binary tree, whose root is a. Prove that the numbers in every line of the...
  22. R

    [number theory] prove that lim x->infinity pi(x)/x=0

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  23. B

    Number Theory: nth root of n is irrational

    1. For n ≥2, n^(1/n) is irrational. Hint provided: Use the fact that 2^n > n2. This is probably familiar to many. By contradiction, n = a^n/b^n --> a^n = n(b^n) --> n|a^n --> n|a Am I trying to force the same contradiction as with 2^1/2 is rational, that is, that a/b are not in lowest terms? Or...
  24. MathWarrior

    Abstract Algebra vs Number Theory?

    I was wondering if one wanted to pursue learning more about cryptography which of these classes would be the most important? Number theory of abstract algebra?
  25. D

    Number Theory: Matrix Exponential

    Homework Statement If I have a matrix M, say 30 5 20 16 How do I calculate M^{1870} mod 101 using Euler's Theorem. Homework Equations I have so far worked out M ^{2} mod 101 to be 91 28 11 53 and thought I could use this as 2x935=1870 The Attempt at a Solution I...
  26. L

    Riemann hypothesis and number theory

    Would the field of the number theory collapse or flourish if the Riemann Hypothesis is proved as true?
  27. P

    Solving Affine Cipher with Number Theory

    Homework Statement Decipher the following text KQEREJEBCPPCJCRKIEACUZBKRVPKRBCIBQCARBJCVFCUPKRIOF KPACUZQEPBKRXPEIIEABDKPBCPFCDCCAFIEABDKPBCPFEQPKAZ BKRHAIBKAPCCIBURCCDKDCCJCIDFUIXPAFFERBICZDFKABICBB ENEFCUPJCVKABPCYDCCDPKBCOCPERKIVKSCPICBRKIJPKABI Homework Equations I know that...
  28. R

    [number theory] product of co primes congruent to 1 (mod m)

    Homework Statement Let b1 through b_phi(m) be integers between 1 and m that are coprime to m. Let B be the product of these integers. Show that B must be congruent to 1 or -1 (mod m) Homework Equations None. The Attempt at a Solution Well, I know that the quantity B appears during the...
  29. S

    Number Theory Puzzler: Proving N-S is a Multiple of 9

    Given: N is a four digit number. S is the sum of N's digits. Prove: N minus S is a multiple of 9.
  30. C

    I have no knowledge of number theory

    hi, i have only basic knowledge of number theory, but would like to know a hell lot, like maths major level or something(especially about fractals). is there any good site where i could? and please don't suggest wikipedia.
  31. M

    Proving the Primality of (2^n)+1: A Number Theory Question

    I need help. I'm trying to prove that if (2^n)+1 is prime, then there exists an integer k>=0 such that n=2^k. If n is odd, then (2^n)+1=(2^(2k+1))-(-1)^(2k+1)=(2+1)(stuff...)=(3)(stuff) so it's not prime, a contradiction. So I've knocked out half of the possible n's. What I'm struggling...
  32. P

    What Repunits are Divisible by Factors of b+1 and b-1?

    Homework Statement A base b repunit is an integer with base b expansion containing all 1's. a) Determine which base b repunits are divisible by factors b-1 b) Determine which base b repunits are divisible by factors b+1 Homework Equations R_{n}=\frac{b^{n}-1}{b-1} The Attempt...
  33. D

    Number theory: confused about the phrase an integer of the form

    Number theory: confused about the phrase "an integer of the form" Homework Statement Prove that any prime of the form 3k+1 is of the form 6k+1 Homework Equations The Attempt at a Solution I'm not sure where to start at all. I tried rewriting 3k+1 as 6k+2=6k+(6-4)=6(k+1)-4. But...
  34. D

    Number theory: simple gcd question

    Homework Statement If ax+by=1, then (a,b)=1. Homework Equations The Attempt at a Solution I am just wondering if this is true. Because I know it is not true if ax+by=c, then (a,b)=c. Here is a proof I came up with: Suppose (a,b)=c, c>1.Then c|a and c|b, but then from our...
  35. R

    [number theory] prove that x^2+y^2=3 has no rational points

    Homework Statement The actual problem is: "Does x^2+y^2=3 have any rational points? If so, find a way to describe all of them. If not, prove it."Homework Equations NoneThe Attempt at a Solution I found a book on Google Books (can't find it again) that said that this circle has no rational...
  36. D

    Number Theory: Simple Divisibility & GCD

    Homework Statement Prove that if N=abc+1, then (N,a)=(N,b)=(N,c)=1. Homework Equations The Attempt at a Solution Assume N=abc+1. We must prove (N,a)=(N,b)=(N,c)=1. Proceeding by contradiction, suppose (N,a)=(N,b)=(N,c)=d such that d\not=1 . Then we know, d | N and d | abc. Thus...
  37. P

    Exploring Number Theory: Modulo 4 and Additive Orders Homework

    Homework Statement 1) What are the possible values of m^{2} + n^{2} modulo 4? 2) Let d_{1}(n) denote the last digit of n (the units digit) a) What are the possible values of d_{1}(n^{2})? b) If d_{1}(n^{2})=d_{1}(m^{2}), how are d_{1}(n) and d_{1}(m) related? 3) a)...
  38. R

    [number theory] find number in certain domain with two prime factorizations

    Homework Statement My domain i numbers of form 4k+1. n divides m is this domain if n=mk for some k in the domain. A number is prime in this domain if its only divisors are 1 and itself. My problem is to find a number in the domain with multiple prime factorizations. Homework Equations...
  39. L

    Product of divisors number theory problem

    Homework Statement prove using induction: for any n =1,2,3... the product of the divisors of n = n^(number of divisors of n (counting 1 and n)/2) Homework Equations The Attempt at a Solution I understand why this is the case, but I'm having trouble with the induction step. if...
  40. L

    Convergence of Sum of Reciprocals for Numbers Starting with Nine

    Homework Statement Does the sum of the reciprocals of natural numbers starting with nine converge? In other words, does Sigma 1/n with n being numbers starting with nine, converge? Homework Equations The Attempt at a Solution I know that subsets of the natural numbers with...
  41. R

    Number theory- prove no three ppt's with same value c

    Homework Statement The problem is that I have to prove that there aren't three or more primitive pythagorean triples with the same value of c. A primitive pythagorean triple has has no values, a, b, or c that have common factors. The actual question is if this is possible, and if not prove it...
  42. K

    Difficult Number Theory question

    Let q be a prime, k= q-2 and X be an Integer, I have found solutions for q=5,7,17 for the equation (2^k) - 7 = X^2 . I have checked q for up to 1000000 but was not able to find any other solutions. Please prove if there can be more solutions or none for this equation. Thanks in...
  43. H

    Is Number Theory useful to physicists

    So I'm stil deciding whether or not I want to do a math/physics major (as opposed to just a physics major), and I was wondering if Number Theory is at all useful to physicists. I ask this because it's the easiest of the three classes I have left for my math major, which would make it perfect...
  44. K

    Number theory - euler's phi function

    Homework Statement Let p = a prime. Show {x}^{2} ≡ a (mod {p}^{2}[/tex]) has 0 solutions if {x}^{2} ≡ a (mod p) has 0 solutions, or 2 solutions if {x}^{2} ≡ a (mod p) has 2. The Attempt at a Solution OK, my mistake, I don't think this has anything to do with the phi function. But I don't...
  45. B

    Introductory number theory textbook

    What is a good introductory textbook for beginners?
  46. arivero

    Is there a probabilistic approach to number theory conjectures?

    For instance, let's say that you want to study fermat x^n+y^n=z^n for n=3; do not mind that we already know the answer :-) We could consider the densities of exact cubes, d(n), and then to calculate joint probabilities for d(Z), d(X) and d(Y). The mechanism can be applied, for instance, to...
  47. A

    Proving the Sum of Odd Numbers in Number Theory Problem | Homework Statement

    Homework Statement Show that for every odd positive integer n the following is correct xn + yn = (x+y)(xn-1 - xn-2y + xn-3y2 - ... - xyn-2 + yn-1) Homework Equations The one above. The Attempt at a Solution I have an idea about using induction to prove this. My idea is to...
  48. L

    How to Prove Common Divisors Divide the G.C.D.?

    Homework Statement Prove that for two integers m,n: all the common divisors divides the g.c.d.(m,n). Homework Equations The Attempt at a Solution g.c.d = aA +bB ; where a, b are the integers and let d be a common divisor, then: d|a and d|b. After this I have no clue where...
  49. M

    How to Solve Congruence Problems with Modulo Arithmetic

    I am on the http://cow.temple.edu/~cow/cgi-bin/manager website working some congruence problems, here you can plug in answers over and over until you get them right. Three problems still baffle me: 1) With Mod24, find the solution of 3-15-21=. Here I just pretended that none of the...
  50. C

    Number Theory with modular arithmetic

    Homework Statement When a is odd, show \frac{a^2-1}{8} is an integer. Then prove by induction n \geq 2 that for all odd numbers a_1,a_2,...,a_n, \frac{(a_1a_2...a_n)^2 - 1}{8} \equiv \frac{a^2_1 - 1}{8} + \frac{a^2_2 - 1}{8} + ... + \frac{a^2_n - 1}{8} \ mod \ 2 Homework Equations The Attempt...
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