What is Integer: Definition and 620 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. L

    MHB Solving Integer Equations for Integers Not Hit By Any Expression

    Hello I am trying to simplify a set of equations to something easier to work with. For example 11x+4 13x+11 31x+17 19x+2 14x+9 Over integer values for x, I need to find numbers that won't be hit by these equations. The first one hits 4,15,26,37,48,59 for x=0,1,2,3,4,5. Is there a way to...
  2. J

    Solid State Books on the Integer Quantum Hall effect

    Hi, does anybody know of any good sources to learn about the Integer Quantum Hall effect from the perspective of theoretical physics? Any suggestion will be appreciated, thanks.
  3. E

    MHB Proving an Infinite Number of Integer Points on a Level Surface

    Given is the function f: R2 -> R, with f(x,y)=x2+y2-6xy+8y. The level surface f(x,y)=1 contains infinitely much points (x,y) where x and y are integer. How can I prove this? I see that it is true with some examples, but how can I prove this. Do I need to use the gradient? Or tangent planes? Or...
  4. Z

    MHB ZyBooks: 2.4.1: Enter the output of the integer expressions.

    I'm not sure how to do these problems here. Help me learn them please!
  5. lfdahl

    MHB (3+√a)^(1/3)+(3-√a)^(1/3) is an integer, find a

    Determine the positive numbers, $a$, such that the sum: $$\sqrt[3]{3+\sqrt{a}}+\sqrt[3]{3-\sqrt{a}}$$ is an integer.
  6. lfdahl

    MHB A finite number of positive integer solutions 1/x+1/y=p/q

    Given a rational number, $\frac{p}{q}$, show that there are only a finite number of positive integer solutions to the equation: $$\frac{1}{x}+\frac{1}{y}=\frac{p}{q}$$
  7. mfb

    A Find positive integer solutions to a/(b+c)+b/(a+c)+c/(a+b)=4

    What an innocently looking equation. If we allow negative integers, a=4, b=-1, c=11 is a solution. Do some tricks with divisibility? Solve for a? Brute force with the computer?It won't help. There are solutions, but the smallest solution has 80 digits. What happens if we replace 4 by other...
  8. lfdahl

    MHB Prove x^31+x^32+x^33+x^34+x^35=n has an integer solution for any integer n.

    Prove, that the equation \[x_1^3+x_2^3+x_3^3+x_4^3+x_5^3 = n\] has an integer solution for any integer $n$.
  9. 6

    B Integer solutions for equations

    I am trying to understand how to find solutions for a problem when parameters are limited to positive integers. Example: 30x+19= 7y+1 =a ; where x,y,a are positive integers Wolframalpha outputs: a = 210 n + 169, x = 7 n + 5, y = 30 n + 24, n element Z(integers) 30*7= 210 (obviously) How do I...
  10. Anshul23

    Highschool graduate dealing with a triple integral?

    I recently came across a problem in Irodov which dealt with the gravitational field strength of a sphere. Took some time to get my head around it and figure how to frame a triple integral, but it felt good at the end. Am I going to start seeing triple integrals in the freshman year tho? If so...
  11. T

    Can this polynomial be factored into two integer products

    Homework Statement Homework Equations none The Attempt at a Solution i assumed it can be factored into the form ## (x^2 + m_1 x + m_0)(x + n_0) ## by comparison of coefficients ## m_0 n_0 = -abc -1\\ m_1 + n_0 = -a -b -c\\ m_0 + m_1 n_0 = ab +ac + bc\\ ## the only other information i have is...
  12. T

    Proving a polynomial cannot be factored with integer coefficients

    Homework Statement [/B]Homework EquationsThe Attempt at a Solution i tried to do it by writing it as ## a_{1999} x^{1999} + a_{1998} x^{1998} ... a_0 \pm1 = 0 ## for 1999 different integer values of x i am thinking of writing it as ## a_{1999} x^{1999} = -a_{1998} x^{1998} - a_{1997} x ^...
  13. A

    A Multiplication and Addition to get an integer

    Hi, to understand finally the Laue equation for diffraction I am missing something : h*p+k*q+l*r = integer. Given that p,q,r are integers how come h,k,l MUST BE INTEGERS as well? Say p=q=r=2, than h=k=l=1/2 works just fine. I understand that there is something about a common...
  14. E

    I Longitudinal resistivity in Integer Quantum Hall Effect

    I have studied the integer quantum hall effect mainly from David Tong's notes and i understand how the ## \rho_{xy}## is quantized in terms of the chern number. What I don't understand is - how the chern numbers relate to the number of filled Landau levels though. - I also don't understand the...
  15. T

    Proving a polynomial has no integer solution

    Homework Statement let p(x) be a polynomial with integer coefficients satisfying p(0) = p(1) = 1999 show that p has no integer zeros Homework EquationsThe Attempt at a Solution ## p(x) = \sum_{i= 0}^{n}{a_i x^i} ##[/B] using the given information a0 = 1999( a prime number) and ## a_n +...
  16. D

    I Integer Cevians - Equilateral Triangles

    Is anybody familiar with any theory of integer cevians on equilateral triangles? More specificaly, I was trying to find something about the number of integer cevians that divide the side in integer parts. Like, the eq triangle of side 8 have cevian 7 dividing one side into 3+5. Only reference...
  17. R

    Is there any way to calculate this integral?

    I have done it by the parametric form of σ, but if I change σ to implicit form that is G(x,y,z)=x^2+y*2+z^2-R^2=0 I don't know how continue. The theory is: where Rxy is the projection of σ in plane xy so it's the circumference x^2+y^2=R^2
  18. kaliprasad

    MHB Infinite Isosceles Triangles w/ Integer Sides & Areas

    Show that there are infiinite isosceles triangles which have integer sides and integer areas
  19. B

    Integer and Rational Number Subtleties in an Algebra Problem

    Homework Statement Let ##S = \{\frac{1}{n} + \mathbb{Z} ~|~ n \in \mathbb{N} \}##. I am trying to show that ##f : \mathbb{N} \rightarrow S## defined by ##f(n) = \frac{1}{n} + \mathbb{Z}## is a bijection. Surjectivity is trivial, but injectivity is a little more involved. Homework EquationsThe...
  20. Albert1

    MHB Is there a solution to $x^2+y^2=1992$ for positive integers $x$ and $y$?

    $x^2+y^2=1992---(A)$ pove $(A)$ has no positive integer solution
  21. lfdahl

    MHB Integer Solutions to x^3+y^3+z^3=2: Proving Infinitely Many

    Prove, that the equation:$x^3+y^3+z^3 = 2$- has infinitely many integer solutions.
  22. Albert1

    MHB Integer Part of $$\sum_{n=1}^{2001}\dfrac{1}{a_n+1}$$

    $$a\,\, sequence:a_1=\dfrac {1}{3}\,\, and \,\, \,\, a_{n+1}=a_n^2+a_n,\,\, n=1,2,3,----2000\\ find \,\, the \,\, integer \,\, part\,\, of :\sum_{n=1}^{2001}\dfrac {1}{a_{n}+1}$$
  23. Mr Davis 97

    I How to prove that 2n=1 has no integer solutions

    This might seem like a very simple problem, because we could just say that the only possible solution is n = 1/2, which is not an integer. But I am curious as to how to prove that there is no solution, with no knowledge of rational numbers, just as we can prove that x^2 = 2 as no rational...
  24. Albert1

    MHB Find Integer Solutions to $k=\dfrac{ab^2-1}{a^2b+1}$

    $a,b\in N$ $k=\dfrac {ab^2-1}{a^2b+1}\,\, \,also \,\,\in N$ find pair(s) of $(a,b)$
  25. S

    MHB Integer Arithmetic for Precise Calculation of Irrational Numbers

    I have authored documents of 40 years of computer software development with a mind to collect them into a publication at some point. They have been built around several software topics but mathemetics is a favorite of mine. I find a point of inspiration and write a piece of software around it...
  26. E

    MATLAB How to Set a Variable in Matlab as an Integer Only?

    Is there a function in Matlab that presets the value of a variable as an integer only? For example I will set the variable 'y' as an integer at the very beginning of the code, and whenever the variable gains a new value it automatically returns an integer value. Thank you in advance.
  27. J

    Fortran Integer overflow on assignment

    Hey guys, I have an assignment on fortran, which basically is supposed to read grades of a class and print them in alphabetical order and wether if the student failed or passed the class. We're supposed to do everything using our basic knowledge of fortran. When I run it it says 'integer...
  28. 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...
  29. ChrisVer

    Integer types (short, long, long long, etc) or more

    I have one pretty basic question... What are the types short, long etc used when creating for example integers used for? Whenever I read the description, it says "it's able to store bigger numbers", but to be honest I don't see any practical use of/visualize such a description... Why would I...
  30. M

    MHB Limit of function containing integer part

    Hey! :o I want to calculate the limit $$\lim_{x\rightarrow \infty}x^{100}\left [\frac{1}{x}\right ]$$ When $x\rightarrow +\infty$ it holds that $0<\frac{1}{x}<1$, or not? (Wondering) If yes, it holds that $\left [\frac{1}{x}\right ]=0$ or not? Then $x^{100}\left [\frac{1}{x}\right ]=0$, and...
  31. Mr Davis 97

    Show that an integer is unique

    Homework Statement Suppose that a and b are odd integers with a ≠ b. Show that there is a unique integer c such that |a - c| = |b - c| Homework EquationsThe Attempt at a Solution What I did was this: Using the definition of absolute value, we have that ##(a - c) = \pm (b - c)##. If we choose...
  32. B

    No Divisors Between an Integer and Twice it

    Homework Statement Suppose that ##x \in \mathbb{N}## is such that ##n < x < 2n##, where ##n## is another natural number. I want to conclude that it is impossible for ##x## to divide ##2n##. Also, is there a way to naturally generalize this to all integers? Homework EquationsThe Attempt at a...
  33. moenste

    KE of alpha particle using integer values of nuclear masses

    Homework Statement (a) Cobalt has only one stable isotope, 59Co. What form of radioactive decay would you expect the isotope 60Co to undergo? Give a reason for your answer. (b) The radioactive nuclei 21084Po emit alpha particles of a single energy, the product nuclei being 20682Pb. (b) (i)...
  34. lfdahl

    MHB Is $a$ an Integer and What is its Modulo 5 Value?

    Let $a = 5^{1000}\cdot sin(1000\alpha)$ , where $sin(\alpha)=\frac{3}{5}$. Prove that $a\in \mathbb{Z},$ and find $a\:\: (mod \:\:5)$.
  35. M

    MHB Prove: Positive Integer n Sum Equation

    prove by induction for all positive integers n: 1+5+9+13+...+(4n-3)= n/2(4n-2) i tried this by trying to prove n/2(4n-2)+ (4(k+1)-3) = k+1/2(4(k+1)-2) but it did not work out for me.
  36. J

    MHB Proof: polynomial with integer solutions

    I am stuck with one proof and I need some help because I don't have any idea how to proceed at this moment. The task says: If f(x) is a polynomial with integer coefficients, and if f(a)=f(b)=f(c)=-1, where a,b,c are three unequal integers, the equation f(x)=0 does not have integer solutions...
  37. S

    A Phi-nth quantum scalar field theories where n is not integer

    Consider a quantum scalar field theory with interaction terms of the form ##\phi^{n}##, where ##n## is not an integer. Where are some examples of physical theories which involve such interaction terms?
  38. T

    MHB Ratio test with an integer power of an in numerator

    I have $$\sum_{n = 1}^{\infty} \frac{2^n}{n^{100}}$$ and I need to find whether it converges or diverges. I can use the ratio test to get: $$\lim_{{n}\to{\infty}} \frac{2^{n + 1}\cdot n^{100}}{2^n \cdot (n + 1)^{100}}$$ But I'm not sure how to get the limit from this. I know the limit of...
  39. Joppy

    MHB Show that is the square of an integer

    I couldn't find this problem anywhere else on the forum so I thought I'd post it. If however, I am duplicating, mods feel free to remove the post :p. No doubt many of you know it already, but I found it quite interesting. Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 +...
  40. S

    Prove by induction that r(r-1)(r+1) is an even integer

    Homework Statement Prove by induction, that when r(r-1)(r+1) is an even integer when r=2,3,4... Homework Equations Prove by induction The Attempt at a Solution I began with the base case r=2, leading 6. Then I proceed with r=3, leading 24. Now if r=k is true, then k(k-1)(k+1) is also true...
  41. kaliprasad

    MHB Square Int x: $x(x+1)(x+7)(x+8)$ is Square Integer

    Find all integers x such that $x(x+1)(x+7)(x+8)$ is square of an integer
  42. Pepper Mint

    Integer partitioning problem - for fun

    I have a user input of 2 integers (m,n) Then my system will generate 1 list of M (m,n < M) integers that start at m and end at Mth integer of value xM. The formula to calculate xM is followed by x_0=m x_M=x_{M-1}+n After the list is generated I randomly delete N (N << M) rows from it and given...
  43. a1call

    I Minimize |n-2^x*3^y| over the integer

    Hi, Is there a way to formulate the solution of minimization of: abs(n-2^x*3^y) Over integers x and y for any given integer n? A numeric example that I found by trial and error is: |6859-2^8*3^3|=53 Thanks in advance.
  44. kaliprasad

    MHB Integer Find x,y for $y^2+2y=x^4+20x^3+104x^2+40x+2003$

    Find all solutions in integers $x,y$ of the equation $y^2+2y= x^4+20x^3+104x^2 + 40x + 2003$
  45. I

    Solution of "polynomial" with integer and fractional powers

    Hello, I have a question regarding "polynomials" that have terms with interger and fractional powers. Homework Statement I want to solve: $$ x+a(x^2-b)^{1/2}+c=0$$ Homework Equations The Attempt at a Solution My approach is to make a change of variable x=f(y) to get a true polynomial (integer...
  46. P

    [Discrete] Prove that |nZ| = |Z| for any postive integer n

    I have been studying discrete mathematics for fun and I am kind of stuck on this bijection problem. 1. Homework Statement I wanted to apologize in advance if i put this homework question in the wrong part of the forums. Discrete Math and much logic math is a computer science type math of...
  47. S

    Number of positive integer solutions for given equation

    Homework Statement Let n be an odd integer ≥ 5. Find the number of triplets (x, y, z) of positive integers which satisfy the equation x + y + 2z = n Homework Equations Do not know The Attempt at a Solution Let n = 2k + 1, k ≥ 2 x + y + 2z = n 2z = 2k + 1 - (x + y) ≤ 2k + 1 - 2 (because x +...
  48. kaliprasad

    MHB Integer Challenge: Proving $2A, A+B, C$ integers for $f(x)=Ax^2+Bx+C$

    Let $f(x) = Ax^2 + Bx +C$ where A,B,C are real numbers. prove that if $f(x)$ is integer for all integers x then $2A, A + B, C$ are integers. prove the converse as well.
  49. S

    B Proof that non-integer root of an integer is irrational

    I have been looking at various proofs of this statement, for example Proof 1 on this page : http://www.cut-the-knot.org/proofs/sq_root.shtml I'd like to know if the following can be considered as a valid and rigorous proof: Given ##y \in \mathbb{Z}##, we are looking for integers m and n ##\in...
  50. Dennis Plews

    B An odd integer series formula?

    A few months ago I posted a simple equation that shows an interesting nexus between the difference between the squares of successive integers and the sums of their roots, viz: Where y = x+1 then (x + y) = (y2 - x2) Recently I expanded this relationship as follows: Where n is any integer and y...
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