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 By the early twentieth century, it had been superseded by "number theory".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

    Number Theory: Divisibility and Prime Factorization

    {SOLVED}Number theory/ divisibility Show that m^2 is divisible by 3 if and only if m is divisible by 3. MY attempt: I assumed that 3k=m for some integers k and m. squared both sides and now get. 3n=m where n=3*(3k^2). Thus 3|m^2 Now the problem is when i assume: 3k=m^2 and need...
  2. D

    How does this imply this (number theory)

    so I have 2^{1990}=(199k+2)^{10} expanding I have. 2^{1990}=2^{10}+10.2^9. (199k)+\frac{10.9}{1.2} 2^8.(199k)^2+...+10.2. (199k)^9+(199K)^{10}-(1) now its clear 199|2^{1990}-2^{10} since I can take 199 out of the RHS. but the book seems to imply that the above equation(1) says...
  3. S

    Has anyone tried Gareth Jones' Elementary Number Theory? What's your review?

    Does anyone have any comments or reviews to share on Gareth Jones' Elementary Number Theory? Is it suitable for an introduction to the subject? If not, what is the recommended book?
  4. W

    Pleas, suggest a advanced+serious Number Theory book

    Hi guys, I want to know about some information of books, I recently finished a Number Theory course by Pommersheim and you know it was pretty easy, and it took for 3 days. Please suggest advanced and serious Number Theory book. + set theory. Thank you in advanced.
  5. D

    Choosing Postgrad Programme: Arithmetic Combinatorics Vs Algebraic Number Theory

    Dear all, I am attending a taught postgrad programme starting next October. I can not decide whether to take the Algebraic Number Theory or the (additive/arithmetic) Combinatorics modules. My choice will determine my PhD route, so it is a choice of career rather than just a choice of...
  6. S

    Brain teasing problem in number theory

    Homework Statement Let X=10000000099 represent an eleven digit no. and let Y be a four digit no. which divides X.Find the sum of the digits of the four digit no. Y. Homework Equations None The Attempt at a Solution I guess I have to factorize X. But it is really difficult to...
  7. H

    Efficient way to solve basic high school level number theory questions?

    I signed up for the coaching service for the GRE and when looked through the questions I struggled with elementary number theory. What's the most efficient way to deal deal with the following kind of questions. 1. Positive integer Z_1 divided by 7 gives a remainder of 5 and Z_2 divided by 4...
  8. T

    What are the prime numbers that satisfy k²=n³+1 when n is not a prime number?

    Homework Statement If k is a prime number find all k that satisfy k²=n³+1 n is not a prime number Homework Equations I really have no idea, use any suitable one The Attempt at a Solution all prime numbers are odd except 2. n must be positive natural number n³ = k² -1 =...
  9. M

    Is Number Theory Worth Studying for Physicists?

    Hello! I am a sophmore physics/math major who will probably be going into Mathematical or Theoretical Physics. My question is should I take Number Theory at some point during my undergrad years? On the one hand, it looks like an interesting/fun class (and I love math :) ) , but I've heard it's...
  10. O

    Background needed to understand number theory research papers.

    Hi, I am a computer science student that has found an interest in mathematics. I am currently exploring number theory, among other fields such as abstract algebra, and have gathered an interest in it after glancing at HAKMEM and Hacker's Delight, as well as learning of its importance in fields...
  11. F

    Exploring Two Number Theory Problems: A Challenge for Mathematicians

    I have wrestled with the following two problems for a couple of hours each and have been successful. Now I am interested in how a more experienced mathematician would go about solving these. I encourage you to look past my lack of latex skills (lol), and do your best with the attachment I...
  12. B

    Least Positive Integer N for 4a+7b Form: Prove Property

    Homework Statement Find the least positive integer N such that every integer n \geq N can be written in the form 4a + 7b, where a,b are non-negative integers. Prove your N has this property Homework Equations The Attempt at a Solution Well, I kind of went about doing trial and error. I...
  13. O

    Differential geometry or number theory (which to take)

    hi, I'm entering my 3rd year of PMAT degree and need to make a choice between differential geometry and number theory. These are both undergrad courses. I am trying to decide which would be more interesting/useful to take. I am planning on going into grad school, so it would be nice to choose a...
  14. F

    Finding the Smallest Possible Value of N in a Simple Number Theory Problem

    Positive integers 30, 72, and N have the property that the product of any two of them is divisible by the third. What is the smallest possible value of N? Note I have not yet taken a Number Theory course. I think I have found the solution using a bit of reasoning and some luck. N=60? I...
  15. F

    Another small number theory proof

    Show that if x, y ∈ Z and 3|x2+z2 then 3|x and 3|z Solution: 3|(x-z)(x+z) => 3|x+z or 3|x-z if 3|x+z then 3|(x+z)2 = x2+z2 +2xz => 3|x2+z2 +2xz - (x2+z2) => 3|2xz => 3|xz so 3|x or 3|z where to go from here? just the argument that if 3|x then 3|x2 and therefore 3 must divide z2 =>...
  16. C

    Solve Ancient Indian Number Theory Problem | Minimum Number of Eggs in a Basket

    Homework Statement i'm sure everyone has seen this: Solve the following ancient Indian problem: If eggs are removed from a basket 2, 3, 4, 5, and 6 at a time, there remain, respectively, 1, 2, 3, 4, and 5 eggs. But if the eggs are removed 7 at a time, no eggs remain. What is the least...
  17. E

    A problem in algebraic number theory

    I'm trying to do the homework for a course I found online. A problem on the first homework goes as follows: Suppose A is an integral domain which is integrally closed in its fraction field K. Suppose q in A is not a square, so that K(sqrt(q)) is a quadratic extension of K. Describe the...
  18. F

    What Does L=F(a) Indicate in Number Theory?

    Wouldn't mind a hint on how to start part iii), thanks. edit: in my notes i have for a similar question: 'L=Q(20.5, 30.5) F=Q(60.5) degree of the min polynomial = 2, because L=F(a) and [L:F]=2' (a = alpha) Could someone clarify what L=F(a) means so I can understand the example...
  19. K

    Number theory: Binary Quadratic Forms

    P.S. I'm not sure where to post this question, in particular I can't find a number theory forum on the coursework section for textbook problems. Please move this thread to the appropriate forum if this is not where it should belong to. Thanks!
  20. N

    Simple Number Theory Proof, Again

    Alright, having problems with this question too. It seems to be the same type of number theory problem, which is the problem. Homework Statement Prove "The square of any integer has the form 4k or 4k+1 for some integer k. Homework Equations definition of even= 2k definition of...
  21. N

    Proving the Multiplication of Even Integers is a Multiple of 4: A Simple Proof

    Homework Statement Use direct proof to prove "The product of any two even integers is a multiple of 4." Homework Equations definition of even is n=2k The Attempt at a Solution My proof is going in circles/getting nowhere. So far I have (shortened): By definition even n=2k...
  22. A

    Number Theory Questions: Proving p and x2 Congruencies

    I have a few questions I am having troubles with. If someone can push me in the right direction that would be awesome. Here are the questions: 1. Prove that the prime divisors, p cannot equal 3, of the integer n2-n+1 have the form 6k+1. (Hint: turn this into a statement about (-3/p) ) 2...
  23. A

    Proving 3 as a Quadratic Non-Residue of Mersenne Primes | Number Theory Problem

    I was just working on some problems from a textbook I own (for fun). I am not sure how to start this problem at all. Here's the question: Show that 3 is a quadratic non-residue of all Mersenne primes greater than 3. I honestly don't know how to start. If I could get some help to push me...
  24. A

    Prove Number Theory Proofs: Sum Irrational, (m+dk) mod d, x^2=x, n^2 mod 3

    1. For any positive integer n, if 7n+4 is even, then n is even. 2.Sum of any two positive irrational numbers is irrational. 3. If m, d, and k are nonnegative integers with d=/=0 then (m+dk) mod d = m mod 4. For all real x, if x^2=x and x=/=1 then x=0 5. If n is an integer not divisible by 3...
  25. F

    Number Theory: Proving a and b in R, a*b = 0 → a = 0 or b = 0

    For i) I said for a in [0,1], then the group of units are = {f in R | f(a) =/= 0} i.e a continuous function f on [0,1] would have a continuous function g on [0,1] such that f.g=1 but the function would have to be g = 1/f, but this wouldn't be continuous if f(a) = 0 for ii) I have to show for...
  26. F

    Finding the units (number theory)

    Homework Statement Prove that the ring R of polynomials with real coefficients (i.e. f(x) = a0 + a1x + ... + anxn, ai real, are elements of R) has only the constant term a0 as the group of units, providing the constant term isn't zero.Homework Equations u is a unit if there exists a v such...
  27. J

    Number Theory divisibility proof

    Homework Statement Prove that for any n \in Z+, the integer (n(n+1)(n+2) + 21) is divisible by 3 Homework Equations A previously proved lemma (see below) The Attempt at a Solution I sort of just need a nudge here. I have a previously proven lemma which states: If d|a and d|b...
  28. H

    Some help with a number theory problem

    Let a,b>1 be integers such that for all n>0 we have a^n-1|b^n-1. Then b is a natural power of a. I can't find a solution.
  29. C

    Converse of Wilson's Theorem Proof, Beginner's Number Theory

    Prove this converse of Wilson’s Theorem: if m > 4 is a composite number then (m − 1)! ≡ 0 (mod m). (Note: This isn’t true for m = 4, so make sure that this fact is reflected in your proof.) My train of thought...: If m is composite, which has a prime factors r and s such that r does not equal...
  30. C

    Elementary Number Theory: Wilson's Theorem

    I am aiming to prove that p is the smallest prime that divides (p-1)!+1. I got the first part of the proof. It pretty much follows from Fermat's Little Theorem/ Wilson's Theorem, but I am stuck on how to prove that p is the smallest prime that divides (p-1)! +1. I am assuming that every...
  31. G

    Is n Prime if Zero Products and Unique Solutions Exist in Modulo n Arithmetic?

    I'm having some trouble addressing the following two questions in a text I am going through: 1. Show that n is a prime number iff whenever a,b ∈ Zn with ab=0, we must have that a=0 or b=0. 2. Show that n is a prime number iff for every a,b,c ∈ Zn satisfying a not =0, and ab=ac, we have...
  32. M

    Can 7 divide 3^(2n+1) + 2^(n+2) in induction for number theory?

    Homework Statement Show 7 divides 3^(2n+1) + 2^(n+2) The Attempt at a Solution Have proved base case K=1 and for the case k+1 I have got ot the point of trying to show 7 divides 9.3^(2k+1) + 2.2^(k+2). Any pointers would be much appreciated. Thanks in advance
  33. C

    Elementary Number Theory Proof, Integral Ideals

    The first part of the problem is as follows: Any nonempty set of integers J that fulfills the following two conditions is called an integral ideal: i) if n and m are in J, then n+m and n-m are in J; and ii) if n is in J and r is an integer, then rn is in J. Let Jm be the set of all integers...
  34. N

    Number theory and theoretical physics

    Hello dear forum member I wanted to know how about the research on this branch of science.Are many people working on or is it a very very small area because it is too difficult . Are researchers in this area seeking new maths in the short what is the general policy direction of their research
  35. S

    Which Book is Best for Self-Studying Number Theory?

    Hi, I was wondering what introductory book to Number Theory for self study would you recommend? I don't need one that is too streched, just one that will give me a flavor of number theory. Thanks in advance!
  36. D

    Number theory - prove if divisible by 2009

    Homework Statement Prove if there exists an integer whose decimal notation contains only 0s and 1s, and which is divisible by 2009. Homework Equations Dirichlet's box principle :confused: The Attempt at a Solution I'm new to number theory, and I'm aware that I do not have the...
  37. A

    Number Theory Books: Find the Right Book for You!

    I'm looking for a good number theory book which doesn't hesitate to talk about the underlying algebra of some of the subject (e.g. using group theory to prove Fermat's Little Theorem and using ring theory to explain the ideas behind the Chinese Remainder Theorem). I'm still an undergraduate, so...
  38. L

    Number Theory proof linked to decimals

    I am trying to understand (I've already seen the rigorous proof in a real analysis class) why exactly rational numbers have periodic decimal expansions. I have basically boiled it down to proving a seemingly simple statement of number theory (I say seemingly because I don't know any number...
  39. T

    Number theory: primitive roots

    Find a primitive root modulo 101. What integers mod 101 are 5th powers? 7th powers? -I tested 2. -2 and 5 are the prime factors dividing phi(101)=100 so i calculated 2^50 is not congruent to 1 mod 101 and 2^20 is not congruent to 1 mod 101. -Therefore 2 is a primitive root modulo 101 I guess...
  40. icystrike

    How Can You Prove This Number Theory Function?

    This is not a homework , i got this question from the internet. Prove:
  41. A

    Number Theory (Finite and Infinite Sets)

    Homework Statement Why is R\Q not countably infinate or denumerable? Given R (Real Number) is not countably infinate or denumerable and Q (rational number) is denumerable. Homework Equations A set is said to be denumberable or countably infinate if there exists a bijestion of N...
  42. H

    Number Theory Homework: Prove ((2^m)-1) Not Divisor of ((2^n)+1)

    Homework Statement If M and N are positive integers >2, prove that ((2^m)-1) is not a divisor of ((2^n)+1) Homework Equations The Attempt at a Solution Is this correct? I use the well-ordering principle. Let T be the set of all M,N positive integers greater than 2 such...
  43. D

    Exploring the Connections between Number Theory and Group Theory

    Hello, I wonder if somebody could point me to a book (preferably), or paper, link, etc. which explores the relations between number theory and group theory. For example, I am (more or less) following Burton's "Elementary Number Theory" and there is no mention of groups. I also have the...
  44. V

    How to Solve a Modular Arithmetic Equation

    Homework Statement compute 59x +15 \equiv 6 mod 811 Homework Equations The Attempt at a Solution 59x \equiv -9 mod 811 I really don't know hoow to do from here.
  45. C

    Number theory. numbers of the form 111 111.

    Homework Statement consider the number m=111...1 with n digits, all ones. Prove that if m is Prime, then n is prime Homework Equations def of congruence. fermat's and euler's theorem. can also use σ(n): the sum of all the positive divisors of n, d(n): the number of positive divisors of n...
  46. C

    Number theory. modulu question.

    Homework Statement x cong 1(mod m^k) implies x^m cong 1(mod m^(k+1)) Homework Equations x cong 1(mod m^k) <=> m^k|x-1 <=> ym^k=x-1 The Attempt at a Solution starting with ym^k=x-1 add one to both sides ym^k+1=x now rise to the power m. (ym^k+1)^m=x^m <=> subtract the 1^m from...
  47. K

    What is the Proof for (p-1)!≡±1 (mod p) in Number Theory?

    Recall the definition of n! (read n factorial"): n! = (n)(n-1)(n-2) ….(2)(1) =∏(k) In both (a) and (b) below, suppose p≥3 is prime. (a) Prove that if x∈ Zpx is a solution to x square ≡1 (mod p), then x ≡±1 (mod p). (b) Prove that (p-1)!≡±1 (mod p) Zpx x shoud be above p a and b looks...
  48. T

    Finding Squares Between Cubes in Elementary Number Theory

    Im really good at number theory but how to show this statement has me stumped! "Show that among the positive integers greater than or equal to 8, between any two cubes there are at least 2 squares"
  49. I

    Modular arith, number theory problem

    Homework Statement Find the number of roots for the equation x^2+1=0 \mod n \: for \: n = 8,9,10,45 Homework Equations The Attempt at a Solution I have no idea where to start. Could someone help me understand?
  50. T

    Simple number theory, divisibility

    Can you help me with this problem: if a^2 divides b^2, show that a divides b. This was a homework question that I had a while ago and it was solved by using the fundamental theorem of arithmetic. I instead tried to solve it with proof by contradiction: a^2 divides b^2 implies a divides...
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