# What is Elementary number theory: Definition and 39 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 By the early twentieth century, it had been superseded by "number theory".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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2. ### 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...
3. ### 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?
4. ### 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?
5. ### 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?
6. ### 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?
7. ### Any resources of questions on Pell's and Di equations?

Are there any resources of questions on the topic of Pell's and non linear Diophantine equations? I am looking for interesting results which are required to be solved. This is for a typical second year undergraduate level. Thanks.
8. ### 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...
9. ### I Can we have modulo a negative number?

Why can't we have modulo negative number? I have never seen this.
10. ### I How do Calculators add fractions with different denominators

How does computer technology add fractions with different denominators?
11. ### I Number of Primes: How Computers Evaluate

How do computers evaluate the number of primes below a given integer?
12. ### I What are the resources for Pythagorean triples?

What is the most motivating way in introduction to Pythagorean triples to undergraduate students? I am looking for an approach that will have an impact. Good interesting or real life examples will help. Is there any resources for this?
13. ### I How to introduce quadratic residues?

What is the most motivating way to introduce quadratic residues? I would like some concrete examples which have an impact. This is for first year undergraduates doing an elementary number theory course. They have done Diophantine equations, solved linear congruences, primitive roots.
14. ### Are there any resources for questions on primitive roots?

Does anyone know of any resources on questions on primitive roots and order of a modulo n? They need to be suitable for elementary number theory course. (These could be interesting results and challenging ones).
15. ### I What is the most motivating way to introduce primitive roots

I am teaching elementary number theory to first year undergraduate students. How do introduce the order of an integer modulo n and primitive roots? How do I make this a motivating topic and are there any applications of this area? I am looking at something which will have an impact.
16. ### Prove that the product of 4 consecutive numbers cannot be a perfect square

Homework Statement n(n+1)(n+2)(n+3) cannot be a square Homework Equations Uniqueness of prime factors for a given number The Attempt at a Solution I'm not sure but I think I've proved a stronger case for how product of consecutive numbers cannot be squares. I don't know whether it is right...
17. ### 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!
18. ### I How to motivate the study of Fermat's Little Theorem

What is the best way to introduce Fermat’s Little Theorem (FLT) to students? What can I use as an opening paragraph which will motivate and have an impact on why students should learn this theorem and what are the applications of FLT? Are there any good resources on this topic?
19. ### I What resources are there for RSA for the layman?

I am looking for any resources which explain the RSA algorithm for the layman. I have found a number of sources but they all tend to end with a morass of technical details. This is for a first year undergraduate course in number theory who have covered some basic work on modular arithmetic.
20. ### Are there prime numbers n for which S=/0?

We have the set:S={1<a<n:gcd(a,n)=1,a^(n-1)=/1(modn)} Are there prime numbers n for which S=/0?After this, are there any composite numbers n for which S=0? (with =/ i mean the 'not equal' and '0' is the empty set) for the first one i know that there are no n prime numbers suh that S to be not...
21. ### Exploring Pell's Equation: Applications and Resources

What is the best way to introduce Pell’s equation on a first elementary number theory course? Are there any practical applications of Pell’s equation? What are the really interesting questions about Pell’s equation? Are there any good resources on Pell’s equation.
22. ### Sum of Two Squares: Intro to Number Theory

Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?
23. ### Lowest Common Multiple - LCM

What is the most motivating way to introduce LCM of two integers on a first elementary number theory course? I am looking for real life examples of LCM which have an impact. I want to be able to explain to students why they need to study this topic.
24. ### Number Theory Book/Books on elementary number theory

Hello all, I probably should have posted this in a math forum but I don't know of any. Can anyone recommend a book/books on elementary number theory with exercises? My math background is not very strong with very little knowledge of set theory so it should be understood by me. We're covering...
25. ### MHB Methods of elementary Number Theory

Hi! (Cool) I am given the following exercise:Try to solve the diophantine equation $x^2+y^2=z^2$ , using methods of elementary Number Theory. So, do I have to write the proof of the theorem: The non-trivial solutions of $x^2+y^2=z^2$ are given by the formulas: x=\pm d(u^2-v^2), y=\pm 2duv...
26. ### Elementary Number Theory - GCD problems and proofs

Problem 1 Suppose ab=cd, where a, b, c d \in N. Prove that a^{2}+b^{2}+c^{2}+d^{2} is composite. Attempt ab=cd suggests that a=xy, b=zt, c=xz. d=yt. xyzt=xzyt. So (xy)^{2}+(zt)^{2}+(xz)^{2}+(yt)^{2}=x^{2}(y^{2}+z^{2})+t^{2}(z^{2}+y^{2})=(x^{2}+t^{2})(z^{2}+y^{2}) Therefore this is...
27. ### Elementary number theory - prove the following statements

This is my first time posting anything on the forum so I apologize if I do anything wrong. I have enrolled myself into elementary number theory thinking we would be taught how to do proofs however it is apparently expected that we already know how to do this. And so since I am a beginner at...
28. ### 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...
29. ### MHB How and what to teach on an 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 also changed in calculus with the advantage of technology. Similarly in linear algebra there was a linear algebra curriculum study group which produced some...
30. ### MHB What to include on a first elementary number theory course?

I have to teach a course in elementary number theory next academic year. What topics should be included on a first course in this area? What is best order of doing things? The students have a minimum background in proof but this is a second year undergraduate module. I am looking for...
31. ### 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...
32. ### 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...
33. ### 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...
34. ### 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...
35. ### 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?
36. ### 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...
37. ### 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...
38. ### 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"
39. ### Elementary number theory in physics?

Does elementary number theory have any applications in physics? If so how?