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 By the early twentieth century, it had been superseded by "number theory".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

    What are the solutions to these number theory equations?

    Homework Statement Solve the following equations positive integers: (i) a!+b!+c!=d! (ii) a!+b!=25*c! (iii)a!=b^2 Homework Equations For the first two one , i have no idea how to begin . But the third one I may use Bertrand's Postulate some where. Could anyone give me some ideas??
  2. matqkks

    Exploring the Fascinating World of Congruent Mathematics

    What the best way to introduce congruences in a number theory course? I am looking for something which will have an impact. What are the really interesting applications of congruent mathematics?
  3. matqkks

    MHB What Are Some Accessible Unsolved Problems in Number Theory for Teenagers?

    What are the most interesting examples of unsolved problems in number theory which an 18 year can understand?
  4. matqkks

    What Are Some Accessible Unsolved Problems in Number Theory for Teens?

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  5. micromass

    Challenge VII: A bit of number theory solved by Boorglar

    This new challenge was suggested by jostpuur. It is rather number theoretic. Assume that q\in \mathbb{Q} is an arbitrary positive rational number. Does there exist a natural number L\in \mathbb{N} such that Lq=99…9900…00 with some amounts of nines and zeros? Prove or find a counterexample.
  6. matqkks

    How and what to teach on a first year elementary number theory course.

    In the late 80’s and early 90’s there was the idea of ‘calculus reform’ and some emphasis and syllabus changed. The order of doing things in calculus also changed with the advantage of technology. Similarly in linear algebra there was a linear algebra curriculum study group which produced some...
  7. matqkks

    MHB How and what to teach on an elementary number theory course.

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  8. matqkks

    MHB What to include on a first elementary number theory course?

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  9. matqkks

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  10. Z

    Algebraic Number Theory, don't understand one step in a given proof

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  11. qspeechc

    Claimed Proof of ABC Conjecture in Number Theory

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  12. T

    Number theory: Modulus and Divisibility problem

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  13. P

    MHB Can You Solve These Number Theory Challenges?

    1)Prove that x,y are positive integers such that $x^2=y^2-9y$, then x=6 or 20. 2) Let p and q be distinct primes. Show that $p^{q-1}+q^{p-1}=1$ (modpq) Hint for 2) Use Fermats little theorem.
  14. E

    How Many Numbers Cannot Be Written as xa+yb?

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  15. M

    Combinatorics/graph theory or number theory books

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  16. M

    What Are the Best Number Theory Books for Various Levels of Expertise?

    Does anyone have suggestions for number theory references? I'm already familiar with elementary number theory and have some algebra background, but I'm not sure what kind of number theory I'm interested in yet. Thanks!
  17. M

    Number Theory non zero natural numbers

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  18. C

    How related is Number Theory to Physics?

    As I've read and been told, you can never know too much mathematics when you study physics, and I think I read it somewhere here. I have also read that cutting-edge theories like M-Theory need most likely a new the invention of a new type of mathematics to be developed. But my question is...
  19. D

    Number Theory Problem: Finding an Integer n for 1+2+3+...+n to End in 13

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  20. J

    Integer Number Theory - n = p + a^2

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  21. W

    Can this number theory problem be solved purely algebraically?

    Let p be a prime number and 1 <= a < p be an integer. Prove that a divides p + 1 if and only if there exist integers m and n such that a/p = 1/m + 1/n My solution: a|p+1 then there exists an integer m such that am = p+1 Dividing by mp a/p = 1/m + 1/mp So if I choose n = mp(which is...
  22. Nono713

    MHB Introductory number theory challenge

    Let $n = pq$ such that $p$ and $q$ are distinct primes. Let $a$ be coprime to $n$. Show that the following holds: $$a^{p^k + q^k} \equiv a^{n^k + 1} \pmod{n} ~ ~ ~ ~ ~ \text{for all} ~ ~ k \in \mathbb{Z}$$
  23. I

    Senior Thesis Topic in Number Theory

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  24. H

    Solving Odd Digit Divisibility by Five using Permutations

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  25. S

    Intro to Number Theory: Best Books for Beginners

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  26. H

    Solving Arithmetic Progression and Combinations: Question on Number Theory

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  27. R

    What are some recommended e-books on number theory?

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  28. Ryuzaki

    What is the value of a+b+c+d+e?

    Homework Statement a, b, c, d and e are distinct integers such that (5-a)(5-b)(5-c)(5-d)(5-e) = 28. What is the value of a+b+c+d+e? Homework Equations N/A The Attempt at a Solution I tried to solve this in the following manner:- On factorizing 28, we get 28 = 2x2x7. So...
  29. T

    Optics vs. Number Theory: Choosing My Classes for the Upcoming Semester

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  30. W

    Number Theory, with a proof discussed in class (not homework)

    Prove that ordda | ordma, when d|m. Some conditions are 1 ≤ d, 1 ≤ m, and gcd(a,d)=1. What I have so far: let x=ordma, which gives us ax\equiv 1 (mod m) \Rightarrow ax=mk+1 for some k\inZ Let m=m'd. Then ax=mk+1=d(m'k)+1
  31. M

    Number Theory (Modular Arithmetic and Perfect Squares)

    Homework Statement If k is an integer, explain why 5k +2 cannot be a perfect square. Homework Equations n/a The Attempt at a Solution I'm in way over my head and not really sure what type of proof I should be using. In my course, we just went over some number theory and modular algebra so...
  32. M

    Number Theory Euclidean Algorithm

    Homework Statement Suppose that u, v ∈ Z and (u,v) = 1. If u | n and v | n, show that uv | n. Show that this is false if (u,v) ≠ 1. Homework Equations a | b if b=ac [b]3. The Attempt at a Solution I understand this putting in numbers for u,v, and n but I don't know how to...
  33. A

    Number theory; primes dividing proof

    Homework Statement Show there exist infinitely many (p, q) pairs, (p ≠ q), s.t. p | 2^{q - 1} - 1 and q | 2^{p - 1} - 1 Homework Equations We are allowed to assume that 2^{β} - 1 is not a prime number or the power of a prime if β is prime. The Attempt at a Solution Using fermat's little...
  34. STEMucator

    Proving the Divisibility of Relatively Prime Integers

    Homework Statement I've got two questions out of my textbook. I'll list both of them and my attempts below. (1) Suppose : a, b, c\in Z, a|c \space \wedge \space b|c.\spaceIf a and b are relatively prime, show ab|c. Show by example that if a and b are not relatively prime then ab does not...
  35. G

    Exploring the Connections Between Number Theory and Physics

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  36. C

    Number Theory. If d=gcd(a,b,c) then d is a linear combination of a,b, and c

    Homework Statement Several of us claimed that if d=gcd(a,b,c) then d is a linear combination of a,b and c, i.e. that d=sa+tb+uc for some integers s,t, and u. That is true, but we only proved the analogous claim for the greatest common divisor of two numbers, i.e. when d=gcd(a,b). We need...
  37. C

    Number Theory. Argue Is not the square of an integer.

    Homework Statement Argue that (17^4)*(5^10)*(3^5) is not the square of an integer. Homework Equations N/A? The Attempt at a Solution Do I break these up, and show that each is not a square? I'm not sure if that would be correct, but sqrt(17^4)=289 * sqrt(5^10)=3125 *...
  38. C

    Number Theory. If d=gcd(a,b) then

    Homework Statement If d=gcd(a,b) show that gcd((a/d),(b/d))=1 Homework Equations N/A? The Attempt at a Solution Basically, I know that I need to show that 1 is a linear combination of a/d and b/d. I'm not exactly sure how to go about this. Dividing by d gives...
  39. G

    Is There a Predictable Pattern in Prime Exponent Series Gaps?

    Professional Help Needed--Elementary Number Theory I will preface this by saying that I have no formal training in Mathematics. I've taken Calc 1 and a couple of Symbolic Logic classes. Forgive me if I butchered any terminology. However, this has been bugging me for a while, and I would like...
  40. romsofia

    Can Number Theory Enhance Our Understanding of Physics?

    This semester I decided to take elementary number theory instead of intro to philosophy. While I so far am enjoying the class, I'm a physics major, and am looking to pursue research in gravity later down the road (only a freshman, so that's far away). The description for the course: This...
  41. A

    Generalizing the Theorem: Number of Solutions to x^k ≡ 1 (mod n)

    Homework Statement Prove that the equation x^k \equiv 1 (mod p) has exactly k solutions if k|p-1. I'm also curious to know if it's possible to generalize this theorem this way: Prove that the equation x^k \equiv 1 (mod n) has exactly k solutions if k|\varphi(n) where \varphi(n) indicates the...
  42. I

    Number theory divisibility question

    Let a, b and c be positive integers such that a^(b+c) = b^c x c Prove that b is a divisor of c, and that c is of the form d^b for some positive integer d. I'm not sure how to solve this question at all, I need some help.
  43. Q

    Number Theory: Unclear Explanation of Divisibility Question

    Hello, The following problem appears in my number theory text: The answer: I have tried to trace the reasoning in reverse. I understand how we get to the finish (by showing that the number is divisible by all of the relatively prime factors of n, but I don't understand how we...
  44. G

    Undergraduate Level Number Theory Text

    Hey guys. Does anyone know of a good undergraduate level textbook on number theory? I have a pretty solid undergraduate level math background but have never had the chance to take a course on this particular topic. If anyone could recommend a textbook that he/she likes, or is widely used at the...
  45. S

    Good theory in number theory for deep understanding

    Does anybody of a good book in number theory for deep understanding of concepts like kiselev in geometry and apostol in calculus ? I have 'introduction to theory of numbers' by I. Niven and H. S. Zuckerman but I feel it is not suitable for my purpose , there is no description of 'why we did...
  46. M

    Another interesting number theory tidbit

    Hello, I was browsing a set of number theory problems, and I came across this one: "Prove that the equation a2+b2=c2+3 has infinitely many solutions in integers." Now, I found out that c must be odd and a and b must be even. So, for some integer n, c=2n+1, so c2+3=4n2+4n+4=4[n2+n+1]...
  47. W

    Number theory find two smallest integers with same remainders

    Homework Statement Find the two smallest positive integers(different) having the remainders 2,3, and 2 when divided by 3,5, and 7 respectively. Homework Equations The Attempt at a Solution I got 23 and 128 as my answer. I tried using number theory where I started with 7x +2 as...
  48. N

    [Number theory] Calculate the Hilbert symbol

    Homework Statement Determine the Hilbert symbol \left( \frac{2,0}{\mathbb F_{25}} \right) where the F denotes the field with 5² elements. Homework Equations \left( \frac{2,0}{\mathbb F_{5}} \right) = -1 The Attempt at a Solution Due to the formula that I put under "relevant equations"...
  49. D

    PARI/GP or comparable number theory program?

    i'm trying to find a version of PARI or PARI/GP or a comparable number theory program that will run on my MacBook Pro. if anyone knows of a site with an updated version or an emulator that will get older versions of PARI working on my computer or newer and better software for testing out long...
  50. N

    [number theory] x²-a = 0 no solution => n not prime

    Homework Statement Define n = 3^{100}+2. Suppose x^2-53 \equiv 0 \mod n has no solution. Prove that n is not prime. Homework Equations / The Attempt at a Solution Well, I suppose that I'll have to prove that some identity which should be true for n prime is not satisfied in the above case...
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