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 By the early twentieth century, it had been superseded by "number theory".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. S

    A couple of Number Theory questions

    1. Find all solutions x (with 0 ≤ x ≤ 96) to the congruence 13x^385 + 73x^304 + x^290 + 10x^193 + 24x^112 + 70x + 76 ≡ 0 (mod 97) I was able to reduce, using Fermat's Little Theorem, to get 97x^16 + x^2 + 93x + 76 ≡ 0 (mod 97), but I don't know how to proceed from there. Is there another trick...
  2. A

    Number Theory Texts: Suggestions & Prerequisites for Undergraduates

    A senior friend of mine who is going to graduate school in mathematics suggested that I try to get at least some exposure to number theory before applying to/attending graduate school. (I'm a freshman undergrad.) Well, I was going to do so anyway, since it's interesting and even applicable, but...
  3. R

    Number theory proof - gcf and lcm

    Homework Statement Prove gcd(lcm(a, b), c) = lcm(gcd(a, c), gcd(b, c)) I've tried coming up with a way to even rewrite it but I'm not really able to do it.
  4. M

    Number theory LMC and GCF equation.

    Homework Statement (E) : 2Lcm(x,y)-5gcf(x,y0=7 Homework Equations 1- Find the possible values of the the number T=gcf(x,y) 2- Solve in N^2 the equation (E). The Attempt at a Solution For number 1 i transformed the equation and I found an equivalence of 2T Ξ 7(mod 5gcf(x,y) is that...
  5. M

    Finding Solutions for LCM and GCD Equations in Number Theory

    Homework Statement Solve in N^2 the following system of equations: 1- gcd(x,y)=7 and Lcm(x,y)=91 2- x+y=24 and Lcm =40 The Attempt at a Solution Let d=gcd(x,y) I said there exists an α and β so that x=dα and y=dβ and gcd(α,β)=1 And after doing some work i reached that α divides αβ=13...
  6. G

    How Does Modular Arithmetic Prove Divisibility by 17 in Number Theory?

    Let x and y be integers. Prove that 2x + 3y is divisible by 17 iff 9x + 5y is divisible by 17. Solution. 17 | (2x + 3y) ⇒ 17 | [13(2x + 3y)], or 17 | (26x + 39y) ⇒ 17 | (9x + 5y), and conversely, 17 | (9x + 5y) ⇒ 17 | [4(9x + 5y)], or 17 | (36x + 20y) ⇒ 17 | (2x + 3y) Could someone please...
  7. M

    Is Number Theory Too Much to Handle with Other Physics Courses?

    I was planning on just taking physics courses next quarter (3, maybe 4 total) but I have an interest in taking a number theory course. Would it be too much to pile on with QM, Relativity, a physics-based math course, and part II of E&M? I know this question is impossible to answer because each...
  8. E

    Number Theory Problem (sums of consecutive squares)

    Homework Statement The sum of two consecutive squares can be a square: for instance, 32 + 42 = 52 (a) Prove that the sum of m consecutive squares cannot be a square for the cases m = 3; 4; 5; 6. (b) Find an example of eleven consecutive squares whose sum is a square. The Attempt at a...
  9. M

    Number theory: ( remainder theorem.)

    Homework Statement A) Find the remainder of 2^n and 3^n when divided by 5. B)Conclude the remainder of 2792^217 when divided by 5. C)solve in N the following : 1) 7^n+1 Ξ 0(mod5) 2) 2^n+3^n Ξ 0(mod5) The Attempt at a SolutionA) I know that for the first two I have to get 2^n=5k+r and...
  10. M

    Number theory: finding integer solution to an equation

    Homework Statement (E): x^2+y^2=6+2xy+3x The Attempt at a Solution x^{2}+y^{2}=6+2xy+3x\Longleftrightarrow x^{2}-2xy-3x+y^{2}=6\Longleftrightarrow x^{2}+x(-2y-3)+y^{2}=6 Any further help to find the answer??
  11. M

    Combinatorial Number Theory Problem

    Hello, I would like to see a solution to the following problem: Let A be a finite collection of natural numbers. Consider the set of the pairwise sums of each of the numbers in A, which I will denote by S(A). For example, if A={2,3,4}, then S(A)={5,6,7}. Prove that if S(A)=S(B) for two...
  12. P

    Number Theory and Euler phi-function

    Homework Statement Let p be prime. Show that p ∤ n, where n is a positive integer, iff \phi(np) = (p-1)\phi(n). Homework Equations Theorem 1: If p is prime, then \phi(p) = p-1. Conversely, if p is a positive integer with \phi(p) = p-1, then p is prime. Theorem 2: Let m and n be...
  13. R

    Number Theory- arithmetic functions

    Problem: Show that for each k, the function σk(n)=Ʃd|n dk is multiplicative. The attempt at a solution: What I know is that I am supposed to use the Lemma which states that if g is a multiplicative function and f(n)=Ʃd|n g(d) for all n, then f is multiplicative. I am just very confused...
  14. T

    Solving Number Theory: Showing Congruence Has Exactly k Distinct Solutions

    Homework Statement http://i43.tinypic.com/fymy3l.jpg question 22.4 (a) Homework Equations The Attempt at a Solution xk=(xp-1)m = (xp-1-1)(1 + xp-1 +x2(p-1) +...+x(m-1)(p-1)) I know that xp-1-1 = 0 mod p has p-1 solutions but I can't make anything from the geometric sum...
  15. M

    Number Theory Perfect Number Proof

    Homework Statement Show that a number of the form 3m5n11k can never be a perfect number. Any ideas?
  16. M

    Solving for Points in 4D Space with Nonnegative Integer Coordinates

    Homework Statement How many points (x1,x2,x3,x4) in the 4-dimensional space with nonnegative integer coordinates satisfy the equation x1 + x2 + x3 + x4 = 10? I'm not sure which method to use to start this problem. Any ideas?
  17. F

    [Number Theory] Finding principal ideals in Z[√-6]

    [Number Theory] Find all the ideals with the element 6 in them in Z[√-5] Edited original question since I have now found the answer (I realize the title is inconsistent on the forum page), instead I am now trying to do part i) here Is it possible to it this way: Or is the structure of the...
  18. T

    Number Theory least divisor of integer is prime number if integer is not prime

    Homework Statement The question is not really a question from a book but rather a statement that it makes : it says " Obviously the least divisor[excluding 1] of an integer a is prime if a itself is not prime." I kind of believe this statement but I'm having trouble proving the general case...
  19. I

    Number Theory Question Possibly related to combinatorics too.

    Homework Statement Prove that a! b! | (a+b)!. Homework Equations Probably some Number Theory Theorem I can't think of at the moment. The Attempt at a Solution Without loss of generality, let a < b. Therefore b! | \Pi _{k=1}^b a+k. Since (a+1) ... (a+b) are b consecutive...
  20. N

    How to learn to like number theory

    Hello, I'm currently taking a course in number theory, and I usually enjoy every branch of pure mathematics, but somehow number theory is not really exciting me. It's hard to pin-point why exactly... Perhaps the following two feelings: - It's hard to see a real structure when trying to tackle a...
  21. Y

    Number theory - quadratic residues

    number theory -- quadratic residues Homework Statement find all incongruent solutions of each quadratic congruence below. X^2\equiv23 mod 77 Homework Equations X^2\equiv11 mod 39 The Attempt at a Solution it is suffices to X^2\equiv23 mod 7, andX^2\equiv23 mod 11, then how to do next?
  22. S

    Differential Equations or Number Theory for Computer Science?

    I'm getting ready to register for classes for the fall. To make a long story short, I might have to take another math class to satisfy a degree requirement, rather than a computer science class. I'm taking Linear Algebra right now. I enjoy it, and it seems to have a lot of practical...
  23. T

    Number Theory: Divisibility Proof

    Homework Statement Show that if p is an odd prime of the form 4k + 3 and a is a positive integer such that 1 < a < p - 1, then p does not divide a^2 + 1 Homework Equations If a divides b, then there exists an integer c such that ac = b. The Attempt at a Solution We have to do this proof by...
  24. L

    On Mersenne Numbers (number theory)

    Homework Statement For a positive integer k, the number M_k = 2^k - 1 is called the kth Mersenne number. Let p be an odd prime, and let q be a prime that divides M_p. a. Explain why you know that q divides 2^{q-1}-1. I have done this already using Euler's theorem, since q prime implies...
  25. J

    Number theory: gcd(a,b)=1 => for any n, gcd(a+bk,n)=1 for some k

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  26. T

    Test Today Quick Number Theory Question

    Test Today...Quick Number Theory Question Let "a" be an odd integer. Prove that a2n (is congruent to) 1 (mod 2n+2) Attempt: By using induction: Base Case of 1 worked. IH: Assume a2k (is congruent to) 1 (mod 2k+2) this can also be written: a2k = 1 + (l) (2k+2) for some "l" IS: a2k+1 =...
  27. T

    Number Theory Question 2: Proving pn | mn Using Prime Factorization

    Let p be a prime and let m and n be natural numbers. Prove that p | mn implies pn | mn. Attempt: Since mn can be written out as a product of primes i.e: p1p2...pn in which p is a divisor. Raising mn means that there would exist pn primes for each factor of m: mn = m1m2...mn =...
  28. T

    Number Theory fundamental theorem of arithemetic

    I have two full questions on some number theory questions I've been working on, I guess my best bet would be to post them separately. 1) Suppose that n is in N (natural numbers), p1,...,pn are distinct primes, and l1,...ln are nonnegative integers. Let m = p1l1p2l2...pnln. Let d be in N such...
  29. A

    Number theory proof trouble: pesty ellipsis

    Homework Statement Prove that x^n - y^n = (x - y) (x^(n - 1) + (x^(n - 2)y + . . . + xy^(n - 2) + y^(n - 1)Homework Equations This is problem 3, section 1-1 from Andrew's "Number Theory," which I'm using for self-study. It follows the section on the "Principle of Mathematical Induction"...
  30. F

    Number theory ideals proof, where am I going wrong?

    I'm trying to prove part iii) So far: Show x irreducible => no y in D-Dx where <x> is a proper subset of <y> Suppose the contrary that x is reducible => x = ay for some a,y in D-Dx => x is an element of <y> => <x> is a subset of <y> By part i) we showed that if <x>=<y> then a must be a...
  31. F

    Calculate the discriminant of a basis [Number Theory]

    Question: The needed proposition and two examples: This is as far as I have got: I need to reduce this (I think) so I can represent is as a matrix! Any idea on how to do this? Thanks
  32. F

    Can someone me understand Norms in number theory?

    Here is a section of examples from my lecture notes. Basically I have NO idea how the lecturer created the matrix Aα, and it's not clear anywhere in the lecture notes. I think it's something to do with complex embeddings but I'm not sure. Does anyone know? I'm sure once I know how...
  33. K

    Number Theory Help: Homework Equation on Prime Generator

    Homework Statement Let p be an odd prime. Show that there exists a\in\mathbb{Z} such that [a]\in\mathbb{Z}^{\times}_{p} is a generator and a^{p-1}=1+cp for some c coprime to p. Homework Equations The Attempt at a Solution I honestly have no idea where to even start with this. Any help will be...
  34. D

    Number Theory WOP: Find Smallest Integer of Form a - bk

    I've just begun number theory and am having a lot of trouble with proofs. I think I am slowly grasping it, but would appreciate some clarification or any tips on the following please. Show that if a and b are positive integers, then there is a smallest positive integer of the form a - bk, k...
  35. Y

    Number theory problem about Fermat 's little theorem

    Homework Statement let n be an integer . Prove the congruence below. n^21 \equiv n mod 30 Homework Equations n^7 \equiv n mod 42 n^13 \equiv n mod 2730 The Attempt at a Solution to prove 30| n^21-n，it suffices to show 2|n^21-n,3|n^21-n,5|n^21-n and how to prove them?
  36. D

    Schools Going to CS grad school for Algebra or Number theory problems in Discrete Math

    I am currently a CS undergrad. my university offers no courses in Abstract algebra or Number theory or Topology or Analysis. recently I have got interested in Number theory in Discrete math course. moreover I was and still am interested in algebra too. but the problem is, can I apply to CS grad...
  37. Y

    Number theory problem divisible

    Homework Statement Prove that n ℂ Z+ is divisible by 3( respectively 9). to show that if and only if the sum of its digits is divisible by 3 Homework Equations The Attempt at a Solution so n= 3q, q>3 that n\equiv0 mod 3 n=X1* 10^n+ x2*10^n-1...Xn so need to...
  38. Y

    Number Theory Problem: Proving (a,b)=1 if a|c and b|c

    Homework Statement a,b,c belong to Z with (a,b)=1. Prove that if a|c and b|c, then ab|c Homework Equations let a1,a2...an, c belong to Zwith a1...an pairwise relatively prime, prove if ai|c for each i, then a1a2...an|c The Attempt at a Solution if a|c, then c=ea, b|c, then c=fb...
  39. M

    (another)interesting number theory problem

    a and b are real numbers such that the sequence{c}n=1--->{infinity} defined by c_n=a^n-b^n contains only integers. Prove that a and b are integers. Mathguy
  40. Y

    Prove the number theory conjecture

    Homework Statement prove or disprove the following conjecture: If n is a positive integar, then n^2 - n +41 is a prime number Homework Equations no, just prove or disprove The Attempt at a Solution I think one possible answer may be there is no factorization for this except...
  41. D

    Number theory or intro to topology for comp sci/math

    I'm pursuing dual degrees in mathematics and computer science with a concentration in scientific computing and am trying to decide whether I should take intro to topology or number theory. Interests in no order are computational complexity, P=NP?, physics engines, graphics engines...
  42. J

    Finding (p-1)(q-1) with Good Precision - Number Theory Doubt

    Given a number pq that is the product of two positive integers p and q, is there any way of finding with good precision, (p-1)(q-1)? Or any approximation at the least? Thanks in advance! :D
  43. S

    Why is the positive value considered the 'normal' state for integers?

    Suppose I think of any integer. In this case, 4. 4 has a negative and positive state -4 and +4. My question is, why is the positive value viewed as the 'normal' state for the number to take? Why isn't there a number 4 that isn't positive nor negative? +4 (4) -4 Why doesn't (4) exist...
  44. A

    Would someone tell me about the importance of number theory?

    well, I've recently found myself interested in the subject, I hadn't studied the subject in high school and I haven't taken the course in university yet but since I've read Herstein's abstract algebra book I have become familiar with some congruence equations and other simple stuff. Right now...
  45. putongren

    Algebraic Number Theory Question

    This is actually a Number Theory question, but requires expertise that doesn't go beyond simple algebra. Homework Statement Show that (1+xy)(1+zy)(1+zx) is a perfect square iff (1 + xy), (1+yz) , and (1+zx) are perfect squares. Homework Equations The Attempt at a Solution I initially tried...
  46. W

    Term structure isomorphic to the usual model/structure of number theory

    Hello, suppose I have a set of sentences Ʃ from the language of number theory ( the usual one ). Then, I extend this to a maximally consistent set of sentences Ʃ' and create a henkin term structure for it ( i.e. as in the popular proof of the completeness theorem ). Can it be true that this...
  47. V

    Number Theory: Define G(n) and show property for any prime p

    Homework Statement Define the numbers G_n = \prod_{k=1}^n (\prod_{j=1}^{k-1}\frac{k}{j}). (a) Show that G_n is an integer, n>1; (b) Show that for each prime p, there are infinitely many n>1 such that p does not divide G_n. Homework Equations The Attempt at a Solution I can see that the...
  48. R

    Number Theory Proof Help: Show p-1 & p^(e-1)|∅(n)

    hey guys been staring at this question for a few days and frustratingly nothing springs to mind. If any1 could offer some direction that would be awsome :) let n be a positive integer, and let p be a prime. Show that if e is a positive integer and p^e|n then p-1|∅(n) and p^(e-1)|∅(n)
  49. E

    Proving simple things (elementary number theory)

    Proving "simple" things (elementary number theory) Hey, I was wondering if I could ask for some help proving simple things in number theory, like divisibility things etc. The kind of stuff that you stare at and say "Duh, properties of real numbers...next?" but maybe don't know how to start as a...
  50. G

    Starting learning number theory

    Hi guys I am really interested in number theory and I want to start studying it,because I am interested in logic of numbers. Do you recommend any book to start with ?
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