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. anemone

    MHB Prove that there is no integer a with P(a)=8.

    Let $P(x)$ be a polynomial with integral coefficients. Suppose that there exist four distinct integers $x_1,\,x_2,\,x_3,\,x_4$ with $P(x_1)=P(x_2)=P(x_3)=P(x_4)=5$. Prove that there is no integer $a$ with $P(a)=8$.
  2. evinda

    MHB Proposition-Set of integer p-adics

    Hello! (Wave) Proposition: "$\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds $\bigcap_{n \in \mathbb{N}_0} p^n \mathbb{Z}_p=0$ and $\mathbb{Z}_p \ p^n\mathbb{Z}_p \cong \mathbb{Z} \ p^n \mathbb{Z}$. Especially $p\mathbb{Z}_p$ is the only...
  3. A

    Why will this function always be an integer?

    f(n) is defined as 11x22x33...xnn Then it seems as if f(n)/(f(r).f(n-r)) is always an integer for 0 < r < n. I tried a few cases. Its true for them. Is it always true? I cannot seem to figure out any ways to prove it.
  4. anemone

    MHB Prove Integer Solution for $x_1^3+x_2^3+x_3^3+x_4^3+x_5^3=k$

    Prove that the equation $x_1^3+x_2^3+x_3^3+x_4^3+x_5^3=k$ has an integer solution for any integer $k$.
  5. john baez

    Building the E10 lattice with integer octonions

    Greg Egan just proved something nice: the E10 lattice, famous in string theory and supergravity, can be described as the lattice of self-adjoint matrices with integral octonions as entries! I'm not sure this result is new, but I've been wanting it for quite a while and haven't seen it...
  6. R

    MHB Proof concerning the greatest integer function

    I'm unsure if this is a calculus or precalculus topic, but it's from a calculus book, so I'm putting it here. (Note \lfloor x \rfloor means the floor of x or the greatest integer less than or equal to x.) Prove that \lfloor x \rfloor +\lfloor y \rfloor \leq \lfloor x+y \rfloor \leq \lfloor x...
  7. T

    Maximizing EEPROM Storage Capacity for Arduino: A Basic Understanding

    Hi everyone, so I was relooking into the external EEPROM problem I had earlier for the Arduino, and am thinking that I am missing even just a basic understanding of it. So I'm using this https://www.sparkfun.com/products/525 EEPROM which should have 256kbits of space. I am hoping to save 4...
  8. M

    Number of unique integer factors

    Homework Statement A number divisible by both 16 and 15 is divisible by at least how many unique integers? Homework Equations Combination formula, nCk = n!/(k!(n-k)!) The Attempt at a Solution I found the prime factorization for 16 and 15, 2x2x2x2x3x5. A number divisible by 16 and 15 must be...
  9. evinda

    MHB Show that number is integer 5-adic, find 5 positions...

    Hello! (Wave) I want to conclude that the number $\frac{1}{2}$ is an integer $5-$ adic and to calculate the first five positions of its powerseries. In order to conclude that $\frac{1}{2}$ is an integer $5-$ adic, do I have to use this definition? Let $p \in \mathbb{P}$. The set of the...
  10. evinda

    MHB Ring of integer p-adic numbers.

    Hey! (Wave) Let the ring of the integer $p$-adic numbers $\mathbb{Z}_p$. Could you explain me the following sentences? (Worried) It is a principal ideal domain. $$$$ The function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p$ is an embedding. (So, $\mathbb{Z}$ is considered $\subseteq...
  11. J

    MHB Is This Permutation Formula an Integer Value?

    Prove by permutations or otherwise $\displaystyle \frac{\left(n!\right)!}{\left(n!\right)^{(n-1)!}}$, where $n\in \mathbb{N}$
  12. J

    MHB Integer Quantity Prove: Permutations/Otherwise n^2! / (n!)^n

    Prove by permutations or otherwise $\displaystyle \frac{\left(n^2\right)!}{\left(n!\right)^n}$, where $n\in \mathbb{N}$
  13. R

    MHB Next integer in this sequence, Challenge

    $\sqrt{\text{mbh}_{29}}$ Challenge: Sn = 3, 293, 7862, 32251, 7105061, 335283445, 12826573186, ?, ?, 44164106654163 S1 through S7 begin an infinite integer sequence, not found in OEIS. 1) Find S8 and S9. 2) Does S10 belong to Sn? 3) If S10 is incorrect, what is the correct value of S10...
  14. kaliprasad

    MHB Can a Triangle with Prime Number Sides Have a Whole Number Area?

    Prove that if the sides of a triangle are prime numbers its area cannot be whole number.
  15. anemone

    MHB Prove an equation has no integer solution

    Let $p,\,q,\,r,\,s$ be positive integers such that $p\ge q\ge r \ge s$. Prove that the equation $x^4-px^3-qx^2-rx-s=0$ has no integer solution.
  16. anemone

    MHB Is (m²)/(m)² always an integer?

    Prove that $\dfrac{m^2!}{(m!)^2}$ is an integer, where $m$ is a positive integer.
  17. anemone

    MHB Prove 2x⁴+2y⁴+2z⁴ is the square of an integer

    The sum of three integers $x,\,y,\,z$ is zero. Show that $2x^4+2y^4+2z^4$ is the square of an integer.
  18. anemone

    MHB Determine all postive integer k

    Determine all positive integers $k$ for which $f(k)>f(k+1)$ where $f(k)=\left\lfloor{\dfrac{k}{\left\lfloor{\sqrt{k}}\right\rfloor}}\right\rfloor$ for $k\in \Bbb{Z}^*$.
  19. anemone

    MHB Find Integer $k$ to Satisfy Sum of Inverse Progression > 2000

    Find an integer $k$ for which $\dfrac{1}{k}+\dfrac{1}{k+1}+\dfrac{1}{k+2}+\cdots+\dfrac{1}{k^2}>2000$.
  20. anemone

    MHB Prove a sum is not the fifth power of any integer

    Suppose $X$ is a number of the form $\displaystyle X=\sum_{k=1}^{60} \epsilon_k \cdot k^{k^k}$, where each $\epsilon_k$ is either 1 or -1. Prove that $X$ is not the fifth power of any integer.
  21. V

    Proof of Infinite Integers: A Base-1 Numeral System

    Mathematicians have long held that infinite integers do NOT exist, but here is a very simple argument that shows that they do exist. A list of positive integers Z+ can be formed in a base-1 numeral system as... 1 11 111 1111 . . . 1111111... Since the set of integers is infinite...
  22. anemone

    MHB Integer solutions Challenge

    Prove that $(-1,\,\pm 1)$, $(0,\,\pm 1)$, $(3,\,\pm 11)$ are the only integers solution for the equation $y^2=x^4+x^3+x^2+x+1$.
  23. B

    Zero Divided By Some Integer n

    I tried to prove this claim, as I require it to finish one of my proofs. By definition, if a,b are integers, with a \ne 0, we say that a divides b if there exists an integer c such that b = ca. So, n divides 0 means that there exists some integer c such that n = c*0 = 0. But this would...
  24. M

    Integer solutions for multiple variable equations

    Obviously it will take some brute-force. But how do I minimize the brute-force needed (optimize)? I know one can solve Diophantine equations and quadratic Diophantine equations. But what if I have something like 10 (any number) of variables? (what if there are no squares, what if there are...
  25. W

    Proof for the greatest integer function inequality

    Can anyone help me prove the greatest integer function inequality- n≤ x <n+1 for some x belongs to R and n is a unique integer this is how I tried to prove it- consider a set S of Real numbers which is bounded below say min(S)=inf(S)=n so n≤x by the property x<inf(S) + h we have...
  26. N

    Why is spin number half integer, especially +1/2,-1/2 for electrons?

    Ok guys, I know this must be pretty basic for but I am new to this section of physics. Anyway, my question is a two-part one, I guess: 1) Why does the spin number get only half integer values in fermions and integer values in bosons, mesons, etc.? 2) How do we conclude that the spin number is...
  27. kaliprasad

    MHB No Integer x for Which $P(x)=14$ Given Four Integer Values of $P(x)=7$

    Show that if a polinomial $P(x)$ with integer coefficients takes the value 7 for four different integer values of x then there is no integer x for which $P(x) = 14$
  28. S

    Any integer = a_0 * 2^n + a_1 * 2^(n – 1) + a_2 * 2^(n – 2) + + a_n

    Homework Statement Problem: Prove that every positive integer P can be expressed uniquely in the form P = a_0 * 2^n + a_1 * 2^(n – 1) + a_2 * 2^(n – 2) + . . . + a_n where the coefficients a_i are either 0 or 1. Solution: Dividing P by 2, we have P/2 = a_0 * 2^(n – 1) + a_1 * 2^(n – 2) + . ...
  29. anemone

    MHB Find the greatest positive integer

    Find the greatest positive integer $x$ such that $x^3+4x^2-15x-18$ is the cube of an integer.
  30. anemone

    MHB Solving the Diophantine Equation $a^4 + 79 + b^4 = 48ab$

    Find all integer solutions $(a,\,b)$ satisfying $a^4+79+b^4=48ab$.
  31. PsychonautQQ

    Finding the Inverse Integer Modulo n

    Homework Statement in mod 35, find the inverse of 13 and use it to solve 13x = 9 gcd(35,13) =1 so the inverse exsists: 35 = 2*13 + 9 13 = 1*9 + 4 9 = 2*4 + 1 4 = 4*1 and then to find the linear combination 1 = 9 - (2*4) = 9 - 2(13-9) = 3*9 - 2*13 = 3* (35 - 2*13) - 2*13 = 3*35 - 8*13 =...
  32. K

    Mathematica How to define functions with integer index in mathematica

    There is a vector ##v_i(t)## (i=1,2,3). How to define the three functions in Mathematica? What about ##t_{ij}(t,\vec{x})##? I am trying to solve my vector and tensor equtions with Mathematica. Analytical solution would be perfect but numerical solution would also be fine. Actually I am not...
  33. C

    Find the Integer N, solution attached

    Homework Statement The expression sin(2°) sin(4°) sin(6°)... sin(90°) is equal to a number of the form (n√5)/2^50 where "n" is an integer. Find n Homework Equations geometric sum: a/ 1-r The Attempt at a Solution I found the solution online but have no idea how they got it...
  34. Albert1

    MHB Prove A_n is an integer for all n in N

    $a,b\in N ,\, and \,\, a>b,\,\, sin \,\theta=\dfrac {2ab}{a^2+b^2}$ (where $0<\theta <\dfrac {\pi}{2}$) $A_n=(a^2+b^2)^nsin \,n\theta$ prove :$A_n$ is an integer for all n $\in N$
  35. kaliprasad

    MHB No Positive Integer Solution for $4xy - x - y = z^2$

    show that the equation $4xy - x - y = z^2$ has no positive integer solution
  36. Sudharaka

    MHB Can You Help Solve This Challenging Integer Expression Problem?

    Hi everyone, :) Here's a question I got from a mailing list in our university. Interesting problem but I sill have no clue on how to solve it. Note that this is a research level question and a solution may or may not exist.
  37. N

    Set of vectors whose coordinates are integer (is a subspace?)

    Homework Statement For a set of vectors in R3, is the set of vectors all of whose coordinates are integers a subspace?The Attempt at a Solution I do not exactly understand if I should be looking for a violation or a universal proof. If x,y, z \in Z then x,y,z can be writted as...
  38. kaliprasad

    MHB Integer Solutions for $4x^2 + 9 y^2 = 72z^2$

    solve in integers x, y, z(parametric form) $4x^2 + 9 y^2 = 72z^2$
  39. S

    Every positive integer except 1 is a multiple of at least one prime.

    Homework Statement The problem (and its solution) are attached in TheProblemAndSolution.jpg. Specifically, I am referring to problem (c). Homework Equations Set theory. Union. Integers. Prime numbers. The Attempt at a Solution I see how we have all multiples of all prime numbers in...
  40. T

    Is there an irrational multiple of another irrational to yield Integer

    This question specifically relates to a numerator of '1'. So if I had the irrational number √75: 1/(x*√75) Could I have some irrational non-transindental value x that would yield a non '1', positive integer while the x value is also less than 1/√75? Caviat being x also can't just be a division...
  41. J

    MHB Bilinear integer equation

    If $x,y$ are integer ordered pair of $2x^2-3xy-2y^2 = 7,$ Then $\left|x+y \right| = $ My Try:: Given $2x^2-3xy-2y^2 = 7\Rightarrow 2x^2-4xy+xy-2y^2 = 7$ So $(2x-y)\cdot (x-2y) = 7.$ Now How can I solve after that Help me Thanks
  42. B

    For every positive integer n there is a unique cyclic group of order n

    Hi, I can't understand why the statement in the title is true. This is what I know so far that is relevant: - A subgroup of a cyclic group G = <g> is cyclic and is <g^k> for some nonnegative integer k. If G is finite (say |G|=n) then k can be chosen so that k divides n, and so order of g^k...
  43. Albert1

    MHB Find Integer Part of A: Math Problem

    $A=(\dfrac{16\times72+17\times73+18\times74+19\times75}{16\times71+17\times72+18\times73+19\times74})\times 150$ find the integer part of A
  44. anemone

    MHB Finding $q$ in a Polynomial with Negative Integer Roots

    If $P(x)=x^4+mx^3+nx^2+px+q$ is a polynomial whose roots are all negative integers, and given that $m+n+p+q=2009$, find $q$.
  45. anemone

    MHB Does the equation $a^2=b^4+b^2+1$ have integer solutions?

    Show that the equation $a^2=b^4+b^2+1$ does not have integer solutions.
  46. anemone

    MHB Finding Integer Solutions to $1998a+1996b+1=ab$

    Find all pairs $(a,\,b)$ of integers such that $1998a+1996b+1=ab$.
  47. Albert1

    MHB Find all integer solutions

    $x,y,z,w $ are all integers if (1):$ w>x>y>z$ and(2) :$2^w+2^x+2^y+2^z=1288\dfrac {1}{4} $ find $x,y,z,w$
  48. anemone

    MHB Solve for Positive Integer Solutions

    Find all values of $(a,\,b)$ where they are positive integers for which $\dfrac{a^2+b^2}{a-b}$ is an integer and divides 1995.
  49. anemone

    MHB Integer Solutions: $(x^2-y^2)^2=1+16y$

    Find all integer solutions of the equation $(x^2-y^2)^2=1+16y$
  50. T

    Prove 5|(3^(3n+1)+2^(n+1)) for every positive integer n.

    "Prove ##5|(3^{3n+1}+2^{n+1})## for every positive integer ##n##." (Exercise 11.8 from Mathematical Proofs: A Transition to Advanced Mathematics 3rd edition by Chartrand, Polimeni & Zhang; pg. 282) I'm having difficulty solving this exercise. My first thought was to use induction but I get...
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