What is Integers: Definition and 472 Discussions

An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold



(

Z

)


{\displaystyle (\mathbb {Z} )}
letter "Z"—standing originally for the German word Zahlen ("numbers").ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.

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  1. Albert1

    MHB Thank you! I'm glad it was helpful.

    $a_1=3,a_2=5757,\,\, a_n=\dfrac {7(a_a+a_2+-------+a_n)}{n}, \,\, (n\geq2)\,\, prove \,\,each\,\, term\,\, of\,\, a_n\,\, is \,\, an \,\, integer$ correction : $a_1=3,a_2=5757,\,\, a_n=\dfrac {7(a_1+a_2+-------+a_{n-1})}{n}, \,\, (n\geq2)\,\, prove \,\,each\,\, term\,\, of\,\, a_n\,\, is \,\...
  2. F

    B Very basic noob question about integers

    hi all so this is a very basic question i think and i feel very bad for tumbling here but still i need to clear this, so from childhood i was taught that the negative numbers are less than positive but now when i am studying limits and functions i came across absolute function and it said |-x| =...
  3. Ventrella

    A Physics and Integer Computation with Eisenstein Integers

    I realize this question may not have an obvious answer, but I am curious: I am using Gaussian and Eisenstein integer domains for geometry research. The Gaussian integers can be described using pairs of rational integers (referring to the real and imaginary dimensions of the complex plane). And...
  4. lfdahl

    MHB Can Anyone Help Crack the Nut on Solving (m,n) Pairs for this Equality?

    Find the pairs of nonnegative integers, $(m,n)$, which obey the equality: \[(m-n)^2(n^2-m) = 4m^2n\] So far, I haven´t found a single pair, but I cannot prove, that the set of solutions is empty. Perhaps, someone can help me to crack this nut?
  5. L

    A Why the Chern numbers (integral of Chern class) are integers?

    I am a physics student trying to self-learn Chern numbers and Chern class. The book I am learning (Nakahara) introduces the total Chern class as an invariant polynomial of local curvature form ##F## ## P(F) = \det (I + t\frac{{iF}}{{2\pi }}) = \sum\limits_{r = 0}^k {{t^r}{P_r}(F)} ## and each...
  6. lfdahl

    MHB Find all integers, such that ....

    Find all integers, $n$, such that the set $\{1,2,3,4, ...,n\}$ can be written as the disjoint union of the subsets, $A$, $B$ and $C$ -whose sums of elements are equal.
  7. lfdahl

    MHB Counting Integers of a Specific Form

    Let $a, b \le 2015$ be positive integers. What is the number of integers of the form: \[ \frac{a^4+b^4}{625}? \]
  8. D

    Proving 5 integers to be pairwise relatively prime

    Homework Statement Let n be an integer. Prove that the integers 6n-1, 6n+1, 6n+2, 6n+3, and 6n+5 are pairwise relatively prime. Homework EquationsThe Attempt at a Solution I tried to prove that the first two integers in the list are relatively prime. (6n-1)-(6n+1)=1 (trying to eliminate...
  9. J

    Are two integers coprime if they are coprime mod(n)?

    Homework Statement Show that out of a set of ten consecutive integers, at least one is coprime to all of the others. Homework Equations Lemma: Out of a set of n consecutive integers, exactly one is divisible by n. (Given). The Attempt at a Solution Let a1, a2...a10 be consecutive integers...
  10. D

    MHB For which integers x,y is (4-6*sqrt(2))^2 = x+y*sqrt(2)?

    Hi All I have the following question. I have reviewed my notes but have not been able to crack this. I tried two different ways, both wrong. First: (4-6*\sqrt2)^2= 16-24*\sqrt2-24*\sqrt2+(36*2) = 88-218*\sqrt2 so, $x=88$ and $y=218$My second method was (4-6*\sqrt2)^2= 4^2-(6*2)=28...
  11. giokrutoi

    Finding All Pairs of Positive Integers for (22016+ 5)m + 22015 = 2n + 1

    Homework Statement (22016+ 5)m + 22015 = 2n + 1[/B] find every n and m pairs as they are positive integersThe Attempt at a Solution (22016+ 5)0 + 22015 = 22015 + 1[/B] so one pair is m= 0 , n = 2015 if m =1 the equation is meaningless if m> 1 so there are really amount of powers that...
  12. DuckAmuck

    I Identify Factorial: Is It Possible?

    Is there a way to identify a factorial without referring to computation of a factorial? For example, is there a way to identify 5040 as a factorial and a way to identify 5050 as not a factorial?
  13. Gjmdp

    Foundations Is it too hard "God Created the Integers"?

    This Stephen Hawking book gathers the most important books along human history. In theory, this book is just divulgative, but, Gäuss, Riemann, Gödel... books included in Hawking's are not meant to be divulgative. So, any of you who read the book(/books if separeted) think it's maybe too hard...
  14. Mr Davis 97

    B Finding the first 40 positive odd integers

    I have the following problem: What is the sum of the first 40 positive odd integers? I look at the solution, and it says that "The sum of the first 40 positive odd integers is ##1 + 3 + 5 + \dotsm + 77 + 79##. And then it goes on with the solution. My question is, how do I find that 79 is...
  15. H

    I Two integers and thus their squares have no common factors

    Integers ##p## and ##q## having no common factors implies ##p^2## and ##q^2## have no common factors. Could you prove this without using the fundamental theorem of arithmetic (every integer greater than 1 either is prime itself or is the product of prime numbers, and that this product is unique...
  16. Q

    Combinatorics, Assigning Occurrence Numbers to Integers

    Homework Statement This problem is from MIT OpenCourseWare- a diagram is attached to clarify certain definitions. I'd like to check my answers. The degree sequence of a simple graph is the weakly decreasing sequence of degrees of its vertices. For example, the degree sequence for the 5-vertex...
  17. R

    B Consecutive integers, each relatively prime to some k

    Hello, Say I have some integer n in some interval such that, gcd(n, k) = gcd(n + 1, k) = 1, for some composite odd integer k >= 9 I want to know if such n exists in that interval. To know that one exists suffices. I have tried to think in terms of modular arithmetic where for all primes in k...
  18. Terry Coates

    Unsolved Mystery: A Diophantine Equation with an Unusual Set of Integers

    Has anyone else spotted an unusual set of three different integers A, B, & C such that A^n + B^n - C^n = A + B - C > 0 (n > 1 and A x B x C > 0) I leave the reader to see if they can find this set, or to ask me what they are.
  19. fatgianlu

    Confidence Intervals for not integers numbers ratio

    Hi, I’m having a problem with a particular case of binomial proportion. I want calculate a confidence Intervals for a binomial proportion for an efficiency. This kind of intervals are usually defined for ratios between integers numbers but in my case I had to subtract from both numerators and...
  20. T

    What lies left of a random number on a line of integers

    When I pick a random number on a number line made out of integers, starting from zero and expanding infinite to the right, what can I say about the position of this random number ? To the right the amount of numbers is infinite. To the left is an amount, a number, so that is finite, but it has...
  21. Alpharup

    How can we prove that integers can be defined as odd or even?

    Iam working through Spivak calculus now. The book defines natural numbers as of form N=1,2,3,4... Iam able to prove that every natural number is either odd or even. How can I extend to Z, integers? In one of the problems, Spivak says we can write any integer of the form 3n, 3n+1, 3n+2.( n is...
  22. T

    MHB Alll Positive Integers proof by contraposition

    For all positive integers $n$, $r$, and $s$, if $rs \le n$ then $r \le\sqrt{n}$ or $s \le \sqrt{n}$ Proof: Suppose $r$ , $s$ and $n$, are integers and $r > \sqrt{n}$ and $ s > \sqrt{e}$. Multiply both sides of the first inequality by $s$. I get $sr > s\sqrt{n} $, but the book gives $rs >...
  23. T

    MHB Proof by contraposition for dividing integers

    For all integers $a$, $b$, and $c$, if $a \nmid bc$, then $a \nmid b$ I need to prove this by contraposition. I get that by definition, $b = ak$ for some integer $k$. But I don't get the following step in the textbook: $bc = (ak)c = a(kc)$ I'm guessing there is something very obvious I'm...
  24. DeldotB

    Compute the G.C.D of two Gaussian Integers

    Homework Statement Hello all I apologize for the triviality of this: Im new to this stuff (its easy but unfamiliar) I was wondering if someone could verify this: Find the G.C.D of a= 14+2i and b=21+26i . a,b \in \mathbb{Z} [ i ] - Gaussian Integers Homework Equations None The Attempt...
  25. T

    Proving 2n ≤ 2^n for All Positive Integers n

    Homework Statement [/B] Show that the statement holds for all positive integers n 2n ≤ 2^n Homework Equations Axiom of induction: 1 ∈ S and k ∈ S ⇒ k + 1 ∈ S The Attempt at a Solution Let S be set of integers 2(1) ≤ 2^1, so S contains 1 k ∈ S, 2k ≤ 2^k I want to show k + 1 ∈ S, 2k +...
  26. B

    Number of squareful integers less than x

    I'm doing the exercises from Introduction to Analytic Number Theory by A.J. Hildebrand (online pdf lecture notes) from Chapter 2: Arithmetic Functions II - Asymptotic Estimates, and some of them leave me stumped... 1. Homework Statement Problem 2.14: Obtain an asymptotic estimate with error...
  27. K

    MHB Properties of gcd's and relatively prime integers

    I am studying this in the context of group/ring theory, ideals etc. So I post it here and not in the number theory section. 6. Suppose gcd(a,b)=1 and c|ab. Prove That there exist integers r and s such that c=rs, r|a, s|b and gcd (r,s)=1. One of my attempts: From gcd(a,b)=1 there exist s',t'...
  28. anemone

    MHB Solve for positive integers

    Solve for positive integers the equation $ab+bc+ca=1+5\sqrt{a^2+b^2+c^2}$.
  29. anemone

    MHB Solving $x^3+y^3+z^3=(x+y+z)^2$ with Positive Integers

    Find all solutions in positive integers $z<y<x$ to the equation $x^3+y^3+z^3=(x+y+z)^2$.
  30. anemone

    MHB Prove the equation has no solution in integers

    Prove that the equation $a^4+b^4+c^4-2a^2b^2-2b^2c^2-2a^2c^2=24$ has no solution in integers $a,\,b,\,c$.
  31. anemone

    MHB Proof that x=0 for Integers with Perfect Square Property

    The integers $x$ and $y$ have the property that for every non-negative integer $n$, the number $2^nx+y$ is a perfect square. Show that $x=0$.
  32. PsychonautQQ

    Equivalence mapping from integers to rationals

    Homework Statement Let * and = be defined by a*b means a - b is an element of the integers and a = b means that a - b is an element of the rationals. Suppose there is a mapping P: (* equivalence classes over the real numbers) --> (= equivalence classes over the real numbers). show that this...
  33. Y

    Simple proof regarding integers

    Homework Statement Show that if m and n are integers such that 4|m2+n2, then 4|mn Homework EquationsThe Attempt at a Solution Since 4 divides m2+n2, then we can say that m2+n2 = 4k, where k is an integer. I haven't done any mathematical proofs of any kind yet, but we were supposed to see if...
  34. evinda

    MHB The set of integers is countable

    Hello! (Smirk) Proposition The set $\mathbb{Z}$ of integers is countable. Proof $\mathbb{Z}$ is an infinite set since $\{ +n: n \in \omega \} \subset \mathbb{Z}$. $$+n= [\langle n, 0 \rangle]=\{ \langle k,l \rangle: k+n=l\}$$ We define the function $f: \omega^2 \to \mathbb{Z}$ with...
  35. evinda

    MHB Why can we define the set of integers using equivalence relations?

    Hi! (Smirk) According to my lecture notes: Constitution of integers Equivalence relation $R$ on $\omega \times \omega$ For $\langle m,n \rangle \in \omega^2$ and $\langle k,l \rangle \in \omega^2$ we say that $\langle m,n \rangle R \langle k,l \rangle$ iff $m+l=n+k$. First Step: $R$ is an...
  36. evinda

    MHB Sorting $m$ Integers in $O(m)$ Time

    Hello! (Smile) How could we show that we can sort $m$ integers with values between $0$ and $m^2-1$ in $O(m)$ time? (Thinking)
  37. M

    MHB Assign integers to the vertices of G

    Hey! :o Let $G=(V,E)$ be a directed acyclic graph. I have to write an algorithm to assign integers to the vertices of $G$ such that if there is a directed edge from vertex $i$ to vertex $j$, then $i$ is less than $j$.Is it maybe as followed?? L ← Set of integers S ← Set of all...
  38. anemone

    MHB Find Integer Pairs $x,y$ with Infinitely Many Solutions

    Find all pairs of integers $x,\,y>3$ such that there exist infinitely many positive integers $k$ for which $\dfrac{k^x+k-1}{k^y+k^2-1}$ is an integer.
  39. anemone

    MHB Find Integer $k$: x^2-x+k Divides x^13+x+90

    Find all integers $k$ for which $x^2-x+k$ divides $x^{13}+x+90$.
  40. H

    Proving the Theorem: p!/[(p-i)! * i] = 1/p for Prime Number p and Integer i

    Prove the following theorem: Theorem For a prime number p and integer i, if 0 < i < p then p!/[(p− i)! * i] * 1/p Not sure how to go about this. I wanted to do a direct proof and this is what I've got so far. let i = p-n then p!/[(p-n)!*(p-n)] but that doesn't exactly prove much.
  41. Math Amateur

    MHB Units of the set of all Eisenstein Integers

    In Chapter 1: "Integral Domains", of Saban Alaca and Kenneth S. Williams' (A&W) book "Introductory Algebraic Number Theory", the set of all Eisenstein integers, \mathbb{Z} + \mathbb{Z} \omega is defined as follows:https://www.physicsforums.com/attachments/3392Then, Exercise 2 on page 23 of A&W...
  42. B

    Prove that if x,y, and z are integers and

    Homework Statement Prove that if x,y, and z are integers and xyz=1, then x=y=z=1 or two equal -1 and the other is 1. 2. Homework Equations The Attempt at a Solution Clearly, if I plug in 1 for each variable, or -1 in for two variables and 1 for the remaining variable, then the equation is...
  43. Math Amateur

    MHB Units of the Gaussian Integers, Z[i]

    In John Stillwell's book: Elements of Number Theory, Chapter 6 concerns the Gaussian integers, \mathbb{Z} = \{ a + bi \ | \ a, b \in \mathbb{Z} \}. Exercise 6.1.1 reads as follows: ------------------------------------------------ "Show that the units of \mathbb{Z} are \ \pm 1, \ \pm i \ ."...
  44. anemone

    MHB Solve $(a^2-b^2)^2=1+16a$ for Integers

    Solve the equation in the set of integers: $(a^2-b^2)^2=1+16a$
  45. anemone

    MHB Counting Integers k with Satisfying Equations Involving Non-Negative Integers

    Find the number of integers $k$ with $1<k<2012$ for which there exist non-negative integers $a,\,b,\,c$ satisfying the equation $a^2(a^2+2c)-b^2(b^2+2c)=k$. ($a,\,b,\,c$ are not necessarily distinct.)
  46. J

    MHB Positive integers ordered pairs (x,y,z)

    Total no. of positive integers ordered pairs of the equation 3^x+3^y+3^z = 7299
  47. M

    MHB Find all three-digit integers.

    Given a three-digit integer $n$ written in its decimal form $\overline{abc}$. Define a function $d(n) := a + b + c + ab + ac + bc + abc$. Find, with proof, all the (three-digit) integers $n$ such that $d(n) = n$.
  48. F

    How many ways to write 4 as the sum of 5 non-negative integers?

    the problem: In how many ways can we write the number 4 as the sum of 5 non-negative integers?I realize this is a generalized combinations problem. I can plug it in using a formula, but I want to understand the logic behind why the generalizaed combination formula works. More specifically, my...
  49. anemone

    MHB Find all non-negative integers (x,y)

    Find all paris $(x,\,y)$ of non-negative integers for which $x^2+2\cdot 3^y=x(2^{y+1}-1)$.
  50. D

    Euclidean Algorithm Gaussian Integers

    Hi, Just wondering when using the Euclidean Algorithm to find gcd of 4+7i and 1+3i. Where does 2 and 2+i come from in the follwoing? 4+7i = 2*(1+3i)+(2+i) 1+3i=(1+i)*(2+i) +0? I know you didvide them to get (5-i)/2 and then take closest Gaussian integer then not sure where to go.
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