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 By the early twentieth century, it had been superseded by "number theory".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

View More On Wikipedia.org
  1. mathics

    Integer triangle with integer heights

    I tried to figure it out by proof by contradiction. I assumed wlog that one side is a prime number, which gave a contradiction, and I also found that such a triangle should be isosceles. Then I assumed wlog that one of the heights is a prime number and I also got a contradiction. Therefore, all...
  2. Charles Kusniec

    Discussion on Astronomical Prime Numbers and Re-evaluating the Primali

    Subject: Discussion on Astronomical Prime Numbers and Re-evaluating the Primality of the Number 1 Dear Members of the Physics Forum, I hope this message finds you well. I've been avidly exploring various discussions on prime numbers and came across an intriguing thread on your forum titled "Is...
  3. DAntanov

    Writing L-functions in Python (or any other language)

    Not many code examples exist for how one would go about writing an L-function. Can anyone give me a step-by-step run down of how to do this and/or link me to relevant resources?
  4. M

    Number Theory What books or materials are needed to understand primitive roots?

    do you know any books, videos, or notes that can help me to understand these topics: 1-primitive roots for primes 2-the existence of primitive roots I am using right now elementary number theory and its application by Kenneth H. Rosen to understand these topics, do you know another source that...
  5. ek81

    Collatz conjecture brought me here

    These are the two tables I developed trying to untangle how the Collatz Conjecture operates, and this certainly helps clear the water a bit
  6. Jiketz

    A Determining rationality of real numbers represented by prime digit sequence

    I would like to know if my answer is correct and if no ,could you correct.But it should be right I hope:
  7. A

    Book recommendations (abstract algebra and number theory)

    Hi, For an engineer who graduated and finished typical Cal A,B,C + Linear Algebra + ODE, what book do you recommend to start reading to be a transition to advanced pure math subjects like abstract algebra and number theory? I did deep google search & concluded that that book supposed to include...
  8. T

    MHB Recreational Number Theory, Unsolved Problem

    Find a perfect power k^m > 1 where k, m, k^m do not contain 2 in their decimal digits, nor do share any decimal digit, no matter if k^m might possibly be expressed in more than one way for some value, e.g. 8^2 = 4^3. I do not know if such an integer exists at all, or how many and how large they...
  9. M

    Please review/verify this proof of assertion (Number Theory)

    Proof: Suppose that all primes except for 3 must have remainder of 1 or 2 when divided by 3. Then we have the form 3p+1 or 3p+2. Note that the product of integers of the form 3p+1 also have the form...
  10. M

    Can anyone please review/verify/check this number theory proof?

    Proof: Suppose for the sake of contradiction that gcd(a, b) \neq 1. Then there exists a prime number k that divides both a+b and ab. Note that k divides either a or b. Since k divides a+b, it follows that k divides b. Thus, this is a...
  11. M

    Number Theory Can anyone recommend me a textbook for Number Theory (Undergraduate)?

    So far, I've taken Ordinary Differential Equations and Introduction to Mathematical Proof. My plan is to take "Introduction To Number Theory" for next semester in Spring 2022. But my professor told me that she won't use a textbook for this class. I was wondering what are some of the good...
  12. M

    Can anyone please check my proof for this number theory problem?

    Please view the picture of my work which I've uploaded.
  13. Leo Liu

    I (Elementary number theory) Understanding congruence and modulus

    In this question, how does the step marked with 1 become the step marked with 2? I can see that the transitivity property of congruence is used, but I don’t know what exactly is going on here. Can someone please explain? Also at which step is Congruence Add and Multiply used? Thanks...
  14. A

    I Can the Housie Game Incorporate Negative and Decimal Numbers?

  15. P

    I Understanding the Results of gcd(x,n)

    Hello, I read something about to which I have a small question of understanding. Before I would like to outline it a little. So we have the primes ##p## and ##q## and we know ##p|x## and ##n = p\cdot q## furthermore it follows that ##p|n## also holds. Now it was said that if ##p|x##, then...
  16. T

    A Unsolved problems in number theory and news about them

    Hello people of physicsforums. Has anyone proved these problems in number theory :Riemann hypothesis, Goldbach conjecture, twin primes conjecture, infinitude of prime numbers? If you want make discussion about them. Thank you for allowing me participate in physicsforums and for wanting to make...
  17. LCSphysicist

    Prove this relation between two numbers (Number theory)

    If (u,v) = 1, prove that (u+v,u-v) is either 1 or 2. Where (,) means: $$ux_1 + vx_2 = 1$$ $$u + v(x_2/x_1) = 1/x_1, u(x_1/x_2) + v = 1/x_2$$ $$u + v = 1/x_1 + 1/x_2 - v x_2/x_1 - u x_1/x_2$$ $$u - v = 1/x_1 - 1/x_2 + u x_1/x_2 - v x_2/x_1$$ Now we can express (u+v,u-v). But i am not sure if...
  18. T

    I Solving congruences and diophantine equations in number theory

    Hello. I do not understand how to solve systems of three or two congruences of one unknown of first order, a congruence of one unknown of second order and a system of diophantine equations of two or three unknowns. Could someone help me by providing examples in these cases? Thank you.
  19. J

    What are the key topics in Advanced Calculus and Algebraic Geometry?

    Hello, I am a very experienced Mathematician with a BSc Honours degree in Mathematics and one year MSc studies in Operational Research in Sussex and London Universities respectively. I am interested in Advanced Calculus, Algebras, Positivity in Algebraic Geometry, The standard Model, and many...
  20. MountEvariste

    MHB Is $p\equiv 3\pmod{4}$ a condition for $\pi$ to be an even permutation?

    Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$
  21. MountEvariste

    MHB Can You Prove This Number Theory Problem Involving Primes and Coprime Numbers?

    1. For $(a,b) = 1$ and prime $p\ne 2$, prove that $\displaystyle \left(a+b, \frac{a^p+b^p}{a+b}\right) = 1$ or $p$.
  22. matqkks

    B Number Theory Image: Hooks for Elementary Courses

    Are there any good images that can be used as a hook for an elementary number theory course?
  23. matqkks

    MHB Number Theory Images for Elementary Course

    Are there any good images that can be used as a hook for an elementary number theory course?
  24. P

    Problem about the sum of the divisors of a number

    I've found that ##N_1## is 1. But it's really tiresome to find them one by one. I also tried to use the equation but couldn't. Please help me out.
  25. matqkks

    I Why certain topics in elementary number theory?

    Why do all elementary number theory courses have the following topics - gcd, linear Diophantine equations, Fundamental Theorem of Arithmetic, factorization, modular arithmetic, Fermat's Little Theorem, Euler's Theorem, primitive roots, quadratic residues and nonlinear Diophantine equations?
  26. matqkks

    MHB Why certain topics in elementary number theory?

    Why do all elementary number theory courses have the following topics - gcd, linear Diophantine equations, Fundamental Theorem of Arithmetic, factorization, modular arithmetic, Fermat's Little Theorem, Euler's Theorem, primitive roots, quadratic residues and nonlinear Diophantine equations?
  27. nomadreid

    N mod m =1 for 1<m<7, N mod 7=0. Find N. (N=301, but how?)

    First, I know the answer: 301. I thought (despite the injunction that the problem is to be done without number theory) that the Chinese Remainder Theorem might be of help (if I would use a subset which contained only relatively primes), but that didn't get very far. I also tried to spot a...
  28. matqkks

    I What to include in an introduction on number theory?

    I am writing an introduction to a first course in elementary number theory. The topics are linear Diophantine equations, modular arithmetic including FLT and Euler's Generalization, quadratic residues and Non - linear Diophantine equations. How can I write an introduction to this showing linkage...
  29. R

    I Generating Irrational Ratios in Wave Simulations

    I am trying to write an algorithm that generates two random numbers in a given interval such that their ratio is an irrational number. I understand that all numbers stored on a computer are rational, so it is not possible to have a truly irrational number in a simulation. So, instead I am...
  30. matqkks

    I What should I say about elementary number theory?

    I need to give an option talk about elementary number theory module. I will discuss how it is study of positive integers particularly the primes and give some cryptography applications. What is a good hook to stipulate in this talk regarding an introduction to elementary number theory?
  31. matqkks

    MHB What should I say about elementary number theory?

    I need to give an option talk about elementary number theory module. I will discuss how it is study of positive integers particularly the primes and give some cryptography applications. What is a good hook to stipulate in this talk regarding an introduction to elementary number theory?
  32. C

    MHB Elementary Number Theory proof

    Dear Everyone, Here is the question: "Prove that if $k$ divides the integers $a$ and $b$, then $k$ divides $as+bt$ for every pair of integers $s$ and $t$ for every pair of integers." The attempted work: Suppose $k$ divides $a$ and $k$ divides $b$, where $a,b\in\mathbb{Z}$. Then, $a=kt$ and...
  33. GlassBones

    Prime factors of odd composites

    Homework Statement Let ##n## be odd and a composite number, prove that all of its prime is at most ##\frac{n}{3} ## Homework Equations Some theorems might help? Any ##n>1## must have a prime factor if n is composite then there is a prime ##p<√n## such that ##p|n## The Attempt at a Solution...
  34. M

    Prime Factors Bounds in a Recurrence

    Let $$g(n)$$ be the numerators of the elements of the recursion $$i(n)=i(n-1)+\frac{1}{i(n-1)}$$ when they are expressed in simplest form, with $$i(0)=1$$. Let $$p$$ be the smallest prime factor of $$g(m)$$. Show that $$p>4m-4$$.Homework Equations Euler's Theorem? The Attempt at a Solution OEIS...
  35. matqkks

    I When was the first time the abbreviation "RSA" appeared?

    <Moderator's note: Edited and URL removed due to potential copyright violation.> When was the first publication of the abbreviation RSA (Rivest, Shamir, Adleman) because it does not appear in an article by Martin Gardner of 1977.
  36. Mr Davis 97

    I Question about a number theory result

    Suppose that ##n,j \in \mathbb{N}##, ##j \in [0, n-1]##, and ##n~|~2j##. Why is it the case that ##j = 0## or ##2j = n##? This is used in a proof of something else, but I am getting tripped up on this part. I know it has to do with the fact that ##j \in [0, n-1]##. Is it because ##n## can't ever...
  37. matqkks

    I Is there a resource which is good questions on quadratic

    I am looking for some interesting questions on quadratic residues. Anyone know of any resources for an elementary number theory course. I am not looking for drill problems to practice but some important results.
  38. Cheesycheese213

    B Fermat's little theorem proof?

    So I was taught that If gcd (a, p) = 1, then ap-1 ≡ 1 (mod p) And then the proof was Lemma: Let p be prime, Let i, j ,k = Integers If gcd (k, p) = 1 and ik ≡ jk (mod p) then i ≡ j (mod p) Main Proof: Consider 1a, 2a, 3a, ..., (p - 1)a Taking mod p is some arrangement of 1, 2, 3, ..., p - 1 Then...
  39. M

    Number theory, primitive pythagorean triples

    Homework Statement Find the known pythagorean triangles with sides of integers lengths, given the area of the pythagorean triangle is 60. Homework Equations Pythagorean triangles are right angled triangles. a primitive pythagorean triple is of the form: x=2mn y=m^2-n^2 z=m^2+n^2 gcd(m,n)=1...
  40. A

    Solve the Mystery: Three Non-Negative Integers & Perfect Powers of 2

    Question: There are three non-negative integers with the following property: If you multiply any two of the numbers and subtract the third number, the result is a perfect power of 2. Find these three numbers that satisfy this property. My attempt: I worked out that the three numbers must be...
  41. Delta31415

    I Question about the Divisor Function/Sums and Project Euler

    So I am kind of lost... I don't really know how to ask this. Project Euler is a website that hosts multiple programming contests and I am interested in this problem https://projecteuler.net/problem=608 but my question isn't truly about this problem but a more solution. I know that the Divisor...
  42. nomadreid

    I No Way to Solve π(x) from Riemann's Zeta Function?

    In the last part of https://en.wikipedia.org/wiki/Riemann_zeta_function#Mellin-type_integrals, I read two expressions of Riemann's zeta function ζ(s) in terms of s and of integrals of the prime-counting function π(x) (the second one using Riemann's prime-counting function J(x) from which, the...
  43. S

    MHB Proving Rationality of Squares of 9th Day 2 ARO 2004/2005 Numbers

    Ten mutually distinct non-zero reals are given such that for any two, either their sum or their product is rational. Prove that squares of all these numbers are rational. I tried using 3 of those numbers - a, b and c. And I checked each of the possible situations but I'm not sure if my maths...
  44. T

    Learning number theory for cryptography

    Hi everyone, I am trying to learn the underlying number theory concepts behind cryptography, and I was wondering if anyone knows of good resources for learning about number theory as applied to cryptography. I was hoping to practice writing proofs as well. Thanks!
  45. T

    Euclidean Algorithm terminates in at most 7x the digits of b

    Homework Statement please see the image Homework Equations I'm not sure if this is relevant: ##r_2 \leq \frac{1}{2}r_1## ... ##r_n \leq (\frac{1}{2})^nr_1## The Attempt at a Solution i have shown that ##r_{i+2} < r_i## by showing the ##r_{i+2} - r_i## is negative, but how do I show that the...
  46. andrewkirk

    Challenge Origami Puzzle Challenge

    RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. 2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is...
  47. Aryamaan Thakur

    Prove that r(r^2-1)(3r+2) is divisible by 24

    Can anyone help me with this divisibility problem. My approach:- 24 = 2*2*2*3 Now, This can be written as (r-1)(r)(r+1)(3r+2) There will be a multiple of 2 and a multiple of 3. But how to prove that there are more multiples of 2. PLEASE REPLY FAST!
  48. daniele1234

    Number Theory Undergraduate Number Theory Book Recommendations

    Hi , everyone! This is my first post/thread/anything on this forum so first I apologise if I slip up or make any mistakes. Anyway, my question is about recommendations for textbooks for Undergraduate Number Theory. So far, I have studied Calculus 1-3 (including things like line integrals...