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 Find the least positive integer

    Find the least positive integer $k$ such that $\displaystyle {2n\choose n}^{\small\dfrac{1}{n}}<k$ for all positive integers $n$.
  2. M

    MHB Show that the number a is not a square of an integer

    Hey! :o I have to show that the number $a=201340168052123987111222893$ is not a square of an integer, without doing calculations.Could I solve this in $\mathbb{Z}_8$? I mean that the number $a$ can be written as followed: $$a=3+9 \cdot 10 +8 \cdot 10^2 + 2 \cdot 10^3+...$$ Since at...
  3. P

    MHB Explaining Integer Equations: Why -1 at the End?

    Can some one explain to my why an integer equation that starts with 1 has a -1 at the end of the equation. example: 1 + 2 + 4 + 8 + 16 ... + 2 ^ N = 2 x ( 2 ^ N ) - 1 Conceptually where does the rule come from that there is a minus at the end of the equation. It starts with an odd number...
  4. anemone

    MHB Square of Integer: Showing Integer's Square

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  5. anemone

    MHB Integer Solutions of $a^{a+b}=b^{12}$ and $b^{b+a}=a^3$

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  6. Seydlitz

    The consequence of divisibility definition in integer

    So I think I've just proven a preposition, where ##0## is divisible by every integer. I prove it from the accepted result that ##a \cdot 0 = 0## for every ##a \in \mathbb{Z}##. From then, we can just multiply the result by the inverse of ##a##, to show that the statement holds for ##0##. That is...
  7. S

    Fortran FORTRAN error array bound is not scalar integer

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  8. Seydlitz

    Alternative Proof to show any integer multiplied with 0 is 0

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  9. anemone

    MHB Is the given quantity an integer?

    Your calculator tells you that the quantity $y=(2\cdot\sqrt[3]{2}+1-\sqrt{12\cdot\sqrt[3]{2}-15})^3$ is approximately an integer. Is $y$ exactly an integer? Justify your answer.
  10. X

    How can I find a solution for c and d for all real integer values?

    $$w = \frac{(ab - d) }{c - a - b}$$ I have to solve the above equation for variables `c` and `d` if `w` can be any number from $$w \in (-\infty, +\infty)$$ If we set `w = 0, then w = 1` we can solve for `c and d` $$0 = ab - d$$ $$d = ab$$ $$c = a + b$$ Now if I can substitute the values to...
  11. anemone

    MHB Prove that a function has no integer roots

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  12. F

    Spinning Particles of 1/2 Integer Spin: Explained

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  13. J

    MHB What is the positive integer value of $n$ if $3^{n} + 81$ is a perfect square?

    If $3^{n} +81$ is a perfect square, find a positive integer value of $n$. My Trail:: When $n\leq 4,$ then easy to know that $3^{n} +81$ is not a perfect square. Now let $\displaystyle n = k +4 (k\in \mathbb{Z^{+}}),$ then $3^{N} +81 = 81 (3^{k} +1).$ So $3^{N} +81$ is a perfect square...
  14. anemone

    MHB Can 20(s - t) Be an Integer in a Quartic Equation with No Real Roots?

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  15. anemone

    MHB Find Nearest Integer to $\dfrac{1}{k^3-2009}$

    Let $k$ be the largest root of $x^4+1-2009x=0$. Find the nearest integer to $\dfrac{1}{k^3-2009}$.
  16. P

    Approximation to an average of integer square roots

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  17. anemone

    MHB What is the largest integer less than or equal to the given expression?

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  18. N

    Interrelationship between power of 2 and integer length

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  19. anemone

    MHB Find the number of positive integer values

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  20. anemone

    MHB Finding $f(1)$ in a Polynomial of Integer Coefficients $\leq$ 4

    Given $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_0, a_a,\cdots,a_n$ are all smaller than 4 and are integer, $a_n \in (0, 1, 2,\cdots)$. Given that $f(4)=2009$, find $f(1)$.
  21. J

    MHB Integer ordered pairs in logarithmic equation

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  22. anemone

    MHB Find all pairs (m,k) of integer solutions

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  23. M

    Integer Sequence: Solve & Generate Terms

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  24. anemone

    MHB Find the smallest positive integer n

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  25. anemone

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  26. Y

    Bessel function of second kind with integer order.

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  27. J

    MHB Total no. of positive integer ordered pairs (n,r)

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  28. hxthanh

    MHB What is the general term for integer sequences satisfying a specific condition?

    Define $\{a_n\}$ is integer sequences (all term are integers) satisfy condition $a_n=a_{n-1}+\left\lfloor\dfrac{n^2-2n+2-a_{n-1}}{n}\right\rfloor $ for $n=1,2,...$ *note: $\left\lfloor x\right\rfloor$ is a greatest integer number less than or equal $x$ Find general term of sequences.
  29. J

    MHB Exploring Integer Ordered Pairs in a Unique Equation

    The no. of positive Integer ordered pair $(a,b)$ in $4^a+4a^2+4 = b^2$
  30. Simon Bridge

    Exploring Oddities of Integer Relationships

    I come across some odd stuff online... ... OK there's a typo for the 5 ... should be ##\small [\sqrt{9}]!-(9/9)## and the one for 7 looks a bit forced... What I'm wondering is if there are other sets that do something like this ... i.e. so for a given integer Z, we can find another integer...
  31. Albert1

    MHB Finding the Value of $f(84)$ in an Integer Function

    $\text{given } :x \in\mathbb{Z}$ $f(x)= \begin{cases}x-3 & x \geq 1000 \\f\big [f(x+5)\big ]& x<1000 \end{cases} $ $\text{find } :\,\, f(84)$
  32. E

    C: warning assignment makes integer from pointer without a cast

    C: "warning assignment makes integer from pointer without a cast" 1. I am trying to assign a color to the variable choice if it is equal to one of the 3 input numbers. if(pred==1) { choice = "RED"; } else if(pred==2) { choice = "GREEN"; } else if(pred==3) { choice =...
  33. anemone

    MHB Find the Smallest Integer Challenge

    Determine the smallest integer that is square and starts with the first four figure 3005. Calculator may be used but solution by computers will not be accepted.(Tongueout)
  34. J

    Proving that every non-negative integer can be expressed in binary

    I'm assuming this is a simple proof? I started thinking about it. It suffices to show that every even number can be written as positive powers of 2 (since every odd number is simply an even number plus 20). So let n = 2k for some non-negative integer k; we need to show that 2k = 2c1 + 4c2 +...
  35. U

    Finding out which prime factors a integer is made up by

    Is this line of thought correct? Please correct me where I´m wrong. Will this way of finding prime factors work when A is any integer? Is there a proof for this or a proof that is closely related? Is there a way to do it that requiers less iterations? It has to be a method that requires...
  36. U

    Multiplying any integer with any prime

    PS. I don't speak or write english very well. I´m doing my best. Is it still ok to post questions? a = any integer b = any prime number a * b = c Is there a proof that "c" isn't made up by any other prime factors than the prime factors that make up "a" (except for "b")?
  37. A

    Can I find a calculator that only calculates unique integer partitions?

    I am curious, are there any calculators that calculate integer partitions with the stipulation that the calculator only calculate unique partitions. For example, if I want to calculate the number of ways to sum to 5 using 3 integers I have the following unique sums: 1 1 3 1 2 2 I would...
  38. anemone

    MHB Find Smallest Positive Integer

    Find the smallest positive integer $n$ such that for every integer $m$ with $0<m<1993$, there exists an integer $k$ for which \frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}.
  39. anemone

    MHB Find Integer Solutions Challenge

    Find all pairs $(p, q)$ of integers such that $1+1996p+1998q=pq$.
  40. caffeinemachine

    MHB Poly. in integer coeff. takes infinitely many integers to composites.

    Let $f(x)$ be a polynomial with integer coefficients. Show that $f(n)$ is composite for infinitely many integers $n$. EDIT: As Bacterius has pointed out we need to assume that $f(x)$ is a non-constant polynomial.
  41. C

    Proof that n is not a power of an integer.

    Homework Statement Prove that n! for n>1 cannot be a square or cube or any other power of an integer. Hint: There is always a prime between n/2 and n if n>3 The Attempt at a Solution Lets assume for contradiction that n!=x^r where x and r are natural numbers and n>3 , so there is some prime...
  42. G

    Fortran Problem setting integer precision in Fortran

    I'm trying to work with big integers but for some reason this program won't compile: program prob003 implicit none integer, parameter :: k32 = selected_int_kind(32) integer(kind=k32) :: num = 600851475143 end program prob003 The file name is prob003.f90 and I'm trying to...
  43. anemone

    MHB Find the integer values of p and q.

    For what integers p and q is where x=\sqrt {29}+\sqrt {89} is a root of the equation x^4+px^2+q=0
  44. H

    Integer sum combinatorics problem

    Question: Given a non-negative integer N, show many sets of non-negative integers (a,b,c,d) satisfy 2a+b+c+d=N Proposed (and roadblocked) solution: Case 1: 2a=0 Then there are \binom{N+2}{2} solutions (easy to prove). Case 2: 2a=2 Then there are \binom{N+2-2}{2} solutions. Case 3: 2a=4...
  45. R

    Discrete distribution taking only non-negative integer values

    I can't seem to wrap my head around the types of sums used in probability theory, and here is a classic example. Section 6.1 of this article: http://en.wikipedia.org/wiki/Expected_value#Discrete_distribution_taking_only_non-negative_integer_values The first line of the proof, what is going...
  46. C

    MIPS assembly programming - converting integer to decimal/binary

    I'm writing a MIPS assembly program and I'm trying to figure out a way to display a given integer in either decimal or binary using only one function with two parameters. The function takes in two parameters: an integer and a base (either '2' or '10'). I'd like to not edit anything other than...
  47. Albert1

    MHB Find Integer $n$ in Range $100$ to $1997$

    $n\in N$ $100\leq n\leq 1997$ $\dfrac{2^n+2}{n} $ also is an integer please find n
  48. S

    Understanding Factorials and Multiplying by an Integer

    Homework Statement Hey all. Not super familiar with using factorials, however, they do pop up occasionally. I understand that n! = 1*2*3*...*n. How do we treat factorial when we are multiplying n by an integer before taking the factorial? I know the answer for expanding (2n)!, however, I do...
  49. S

    Integer Solutions to Linear Systems

    Homework Statement We were asked to solve it using Augmented matrix. I just need one more equation though. A jar of coins contains only dimes, nickels, and pennies. There are 98 coins in the jar, and the total value of the coins is $6.49. Set up the system of equations representing this...
  50. anemone

    MHB Find all positive integer solutions of the given equation.

    Find all positive integer solutions of the equation $4x^3-12x^2+5x-10y+36y^2-18y^3+4x^2y+6xy-15xy^2=0$.
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