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. chwala

    Understanding the given proof of integers - Ring theory

    My interest is on the highlighted part ... Now to my question, in what cases do we have ##mn<(m,n)[m,n]?## I was able to use my example say, Let ##m=24## and ##n=30## for example, then ##[m,n]=120## and ##(m,n)=6## in this case we can verify that, ##720=6⋅120## implying that, ##mn≤...
  2. chwala

    Find the GCD of the given complex numbers (Gaussian Integers)

    Hello guys, I am able to follow the working...but i needed some clarification. By rounding to the nearest integer...did they mean? ##z=1.2-1.4i## is rounded down to ##z=1-i##? I can see from here they came up with simultaneous equation i.e ##(1-i) + (x+iy) = \dfrac{6}{5} - \dfrac{7i}{5}## to...
  3. M

    I Is this the correct way to quantify these integers?

    Does the above quantifier represent/symbolize that all of the integers ## a, b, c, d ## cannot be ## 0 ##? Is this correct?
  4. M

    I Planck’s constant: Why are atom vibration frequencies integers only?

    Given the definition of whole numbers as integers, https://www.google.com/search?q=what+is+a+whole+number&rlz=1C1VDKB_en-GB&oq=what+is+a+whole+number&aqs=chrome..69i57j0i512l9.11619j0j15&sourceid=chrome&ie=UTF-8 Is it known why atom vibrations are only at whole numbers ( ref plank’s constant)...
  5. PhysicsRock

    Prove relation between the group of integers and a subgroup

    So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt. $$ 0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...
  6. anemone

    POTW Integer n: Solving the Equation ##\dfrac{n^2+3}{2n+4}## for Integers

    Find all integers n such that ##\dfrac{n^2+3}{2n+4}## is an integer as well.
  7. S

    I Union of Prime Numbers & Non-Powers of Integers: Usage & Contexts

    Is there a name for the union of {prime numbers} and {integers that are not powers of integers}? For example, we would include 2, 3, 5, 7, 11... And also 6, 10, 12... But we exclude 2^n, 3^n, ... and 6^n , 10^n , etc. What are some interesting contexts where this set crops up?
  8. brotherbobby

    Set properties of even (##E##) and odd (##I##) integers

    Problem Statement : I copy and paste the problem as it appears in the text (Lang, Basic Mathematics, 1971). Attempt : There are several questions in both a) and b) above. I type out the question and my answer each time. a) (i) Show that addition for ##E## and ##I## is associative and...
  9. M

    Determine all integers ## n ##

    Observe that ## \phi(1)=\phi(2)=1 ##. This implies ## \phi(1)\mid 1 ## and ## \phi(2)\mid 2 ##. Thus ## n=1 ##. Let ## n=p_{r}^{k_{1}}\dotsb p_{s}^{k_{s}} ## be the prime factorization of ## n ##. Then ## \phi(n)=n\prod_{p\mid n} (1-\frac{1}{p}) ##. Suppose ## \phi(n)\mid n ##. Then ##...
  10. M

    Use the Euclidean algorithm to find integers ## a, b, c ##

    Let ## a, b, c ## and ## d ## be integers such that ## 225a+360b+432c+480d=3 ##. Then ## 75a+120b+144c+160d=1 ##. Applying the Euclidean algorithm produces: ## gcd(75, 120)=15, gcd(120, 144)=24 ## and ## gcd(144, 160)=16 ##. Now we see that ## 15x+24y+16z=1 ##. By Euclidean algorithm, we have...
  11. M

    Determine all integers ## n ## for which ## \phi(n)=16 ##

    Suppose that ## n=p_{1}^{k_1}p_{2}^{k_2}\dotsb p_{r}^{k_r} ## satisfies ## \phi(n)=k ##. Then ## n=\frac{k}{\prod(p_{i}-1)}\prod p_{i} ##. Note that the integers ## d_{i}=p_{i}-1 ## can be determined by the conditions ## (1) d_{i}\mid k, (2) d_{i}+1 ## is prime, and ## (3) \frac{k}{\prod d_{i}}...
  12. anemone

    POTW Find Triplets of Positive Integers with Sum of Cubes

    Find all triples (a, b, c) of positive integers such that ##a^3+b^3+c^3=(abc)^2##.
  13. M

    How to obtain three consecutive integers?

    Let ## a, a+1 ## and ## a+2 ## be the three consecutive integers. Then \begin{align*} &5^{2}\mid a\implies a\equiv 0\pmod {25}\\ &3^{3}\mid (a+1)\implies a+1\equiv 0\pmod {27}\implies a\equiv 26\pmod {9}\\ &2^{4}\mid (a+2)\implies a+2\equiv 0\pmod {16}\implies a\equiv 14\pmod {16}.\\...
  14. D

    I Primes -- Probability that the sum of two random integers is Prime

    im thinking i should just integrate (binominal distribution 1-2000 * prime probability function) and divide by integral of bin. distr. 1-2000. note that I am looking for a novel proof, not just some brute force calculation. (this isn't homework, I am just curious.)
  15. M

    One-line Python Code that returns a sequence of integers

    Hey! 😊 I want to write a one-line Python generator or iterator expression that returns the sequence of integers generated by repeatedly adding the ascii values of each letter in the word “Close” to itself. The first 10 integers in this sequence are: 67, 175, 286, 401, 502, 569, 677, 788, 903...
  16. M

    Proof: Divisibility of Integers by 4

    Proof: Let ## N ## be an integer. Then ## N=a_{m}10^{m}+a_{m-1}10^{m-1}+\dotsb +a_{1}10+a_{0} ## for ## 0\leq a_{k}\leq 9 ##. Note that ## 10^{k}\equiv 0\pmod {4} ## for ## k\geq 2 ##. Thus ## 4\mid N\Leftrightarrow N\equiv 0\pmod {4}\Leftrightarrow a_{1}10+a_{0}\equiv 0\pmod {4} ##...
  17. M

    Divisibility of Integers ## 176521221 ## & ## 149235678 ## by 9 & 11

    First, consider the integer ## 176521221 ##. Observe that ## 1+7+6+5+2+1+2+2+1=27 ##. Since ## 9\mid (1+7+6+5+2+1+2+2+1) ##, it follows that ## 9\mid 176521221 ##. Note that ## 1-2+2-1+2-5+6-7+1=-3 ##. This means ## 11\nmid (1-2+2-1+2-5+6-7+1) ##. Thus ## 11\nmid 176521221 ##. Therefore, the...
  18. M

    Any one of the integers ## 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ## can occur?

    Proof: Let ## a ## be any integer. Then ## a\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8 ##, or ## 9\pmod {10} ##. Thus ## a^{3}\equiv 0, 1, 8, 27, 64, 125, 216, 343, 512 ##, or ## 729\pmod {10} ##. Therefore, anyone of the integers ## 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ## can occur as the units digit of ##...
  19. M

    Prove that the integers ## c, c+a, c+2a, c+3a, ...., c+(n-1)a ## ....

    Proof: Suppose for the sake of contradiction that ## r, s\in {0, ..., n-1} ## for ## r<s ## where ## c+ra\equiv c+sa\pmod {n} ##. Then ## c+ra\equiv c+sa\pmod {n}\implies ra\equiv sa\pmod {n}\implies r\equiv s\pmod {n} ##. Thus ## n\mid (r-s)\implies n<r-s ##, which is a contradiction because...
  20. e2m2a

    A Prime Number Powers of Integers and Fermat's Last Theorem

    From my research I have found that since Fermat proved his last theorem for the n=4 case, one only needs to prove his theorem for the case where n=odd prime where c^n = a^n + b^n. But I am not clear on some points relating to this. For example, what if we have the term (c^x)^p, where c is an...
  21. M

    For n>3, show that the integers n, n+2, n+4 cannot all be prime

    Proof: Let ## n>3 ## be an integer. Applying the Division Algorithm produces: ## n=3q+r ## for ## 0\leq r< 3 ##, where there exist unique integers ## q ## and ## r ##. Suppose ## n ## is prime. Then ## n=3q+1 ## or ## n=3q+2 ##, because ## n\neq 3q ##. Now we consider two cases. Case #1...
  22. M

    Verify that the integers 1949 and 1951 are twin primes

    Proof: Consider all primes ## p\leq \sqrt{1949} \leq 43 ## and ## q\leq \sqrt{1951} \leq 43 ##. Then we have ## p\nmid 1949 ## and ## q\nmid 1951 ## for all ## p\leq 43 ##. Thus, ## 1949 ## and ## 1951 ## are both primes. By definition, twin primes are two prime numbers whose difference is ## 2...
  23. M

    Find the prime factorization of the integers 1234, 10140, and 36000?

    ## 1234=2\cdot 617 ## ## 10140=2\cdot 2\cdot 3\cdot 5\cdot 13\cdot 13 ## ## 36000=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 5\cdot 5\cdot 5\cdot ## Are the answers above correct? Or do I need to put them in canonical form as below? ## 1234=2\cdot 617 ## ## 10140=2^{2}\cdot 3\cdot 5\cdot...
  24. arivero

    I Can integers be defined as N[[sqrt(1)]]?

    Sometimes I have seen a process to build integers and rationals via a sort of Grothendieck product, Z being classes of equivalence in N x N, and Q being classes of equivalence in Z x Z. Now, I was wondering if it makes sense to consider the integers as the extension of ##\mathbb{N}## by ##\sqrt...
  25. Andrew1235

    Finding all subsets of a list of positive integers using backtracking

    The following Python 3 code is provided as the solution to this problem (https://leetcode.com/problems/subsets/solution/) that asks to find all subsets of a list of integers. For example, for the list below the output is [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]. I am not familiar...
  26. J

    A Number of unequal integers with sum S

    Hello, I've been trying to solve this problem for a while, and I found a technical solution which is too computationally intensive for large numbers, I am trying to solve the problem using Combinatorics instead. Given a set of integers 1, 2, 3, ..., 50 for example, where R=50 is the maximum...
  27. docnet

    Does there exist a surjection from the integers to the naturals?

    a) Yes. One surjection from ##\mathbb{Z}## to ##\mathbb{N}## is the double cover of ##\mathbb{N}## induced by ##f:\mathbb{Z}\longmapsto\mathbb{N}## with $$f(z)=\begin{cases} -z & ,\forall z<0\\ z+1 & ,\forall 0\leq z \end{cases}$$ b) Yes. One surjection from ##\mathbb{R}## to ##\mathbb{N}## is...
  28. A

    B Calculating the coprime probability of two integers in a different way

    The probability that two randomly chosen integers to be coprimes is known to be equal to ## \prod_{2}^{\infty}(1-\frac{1}{p^2})=6/\pi^2## I tried to conceptualize the problem in the following way but got different results. Suppose that we pick up an integer at random which could be either prime...
  29. C

    MHB How many times should the value of inputCount be read?

    First, read in an input value for variable inputCount. Then, read inputCount integers from input and output each integer on a newline after the string "value - ". Ex: If the input is 2 25 55, the output is: value - 25 value - 55 Code: #include <iostream> using namespace std; int main() {...
  30. M

    MHB Representation of signed integers of base B

    Hey! :giggle: Consider a representation of signed integers of base $B$, in which the digits are listed in descending order of importance, with the least significant digit corresponding to a positive, and the next digits to an alternate negative and positive value. Thus, a number of this...
  31. Arman777

    Understanding the binary transformation of strings and integers

    For fun, I have decided to implement a simple XOR encryption algorithm. The first step is to convert messages into bytes to perform XOR operation on each bit. The problem has started here. For instance, I want to encrypt this message. I hiked 24 miles. Now I need to turn this text into binary...
  32. maistral

    A Optimization with integers as results

    Say for example I have a dataset (X, Y) which I need to fit to the function y = Ax^2 + By + Cxy. How do I retrieve values of A, B, and C such that they can only be integers? As of now I'm doing grid search which is so taxing.
  33. C

    I Number of Integers (<N) divisible only by one power of 2

    Hi, The original problem was : for a given number k = d + n/d, where d is a divisor of another number n, how many k <= N are prime? When I looked at this problem, for k to be prime > 2, it has to be odd. This implies d and n/d can't both be even or odd. If d = 2, then d is even and n/d has...
  34. A

    Symbol for Integers: Z or I or both?

    Personal Question: Internet says the standardized math symbol for integers is ## \mathbb {Z}##. However, my Alberta MathPower 10 (Western Edition) textbook from 1998 says the symbol is I. I'm guessing that textbook is wrong? Or are both answers correct?
  35. PeroK

    What happens when large integers are involved?

    I've written a program to factorise large numbers (although not that large). The following arithmetic operation goes wrong: x= int(912_321_155_211_368_155/(5)) The result is 182_464_231_042_273_632 Which is clearly not right (should end in 631). The maximum integer on the 64-bit version is...
  36. B

    I Cryptology - Fast Factoring Integers by SVP Algorithms "destroys RSA"

    The summary abstract (describes the method) and full paper are linked. Summary abstract https://eprint.iacr.org/2021/232.pdf
  37. DanjoJojo

    Comp Sci Writing a Little Man Computer program to compare 2 integers

    I've attempted and I can retrieve a 0 when the first input is greater than the second using BRP. I can't seem to get the output of 1 if the first input is equal to the second input. I also don't get a 2 output with the first input being less than the second. I also have no idea how to loop a...
  38. brotherbobby

    To prove that a given quadratic has integral roots

    Given : The quadratic equation ##x^2+px+q = 0## with coefficients ##p,q \in \mathbb{Z}##, that is positive or negative integers. Also the roots of the equation ##\alpha, \beta \in \mathbb{Q}##, that is they are rational numbers. To prove that ##\boxed{\alpha,\beta \in \mathbb{Z}}##, i.e. the...
  39. LCSphysicist

    B What is the identity element in the group {2,4,6,8} under multiplication mod 10?

    Maybe my problem is misunderstand the concept of " a modulo n ". I would appreciate any help to get this concept and understand the grou´p
  40. A

    I Base Pi Integers: Isomorphism with Rationals?

    A proper number is expressed in \pi in a similar way as a decimal integer is expressed in base 2. For example, 4375_{\pi} = 4{\pi^3}+3{\pi^2}+7{\pi}+5. The only exception I make is that the 10 digits are included when expressing a number with \pi. To clarify, the first positive such numbers are...
  41. anemone

    MHB Prove that the sum of 6 positive integers is a composite number

    Let $a,\,b,\,c,\,d,\,e,\,f$ be positive integers and $S=a+b+c+d+e+f$. Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$, prove that $S$ is composite.
  42. anemone

    MHB Positive Integers: Evaluate $a+b+c$

    $a,\,b$ and $c$ are positive integers that satisfy the inequality $ab+3b+2c>a^2+b^2+c^2+3$. Evaluate $a+b+c$.
  43. Svein

    Wanted: A calculator that can handle really big integers and fractions

    I am trying to get one step further with my search for \sum_{n=1}^{\infty}\frac{1}{n^{2s+1}} . Part of the way is to calculate some algebraic expressions containing fractions with really huge numbers (as in (\frac{1}{5^{9}}+\frac{1}{7^{9}}-\frac{1}{17^{9}}-\frac{1}{19^{9}})\div...
  44. D

    I What are all the positive integers that satisfy this equation?

    X^3 + y^2 - z = z^3 - x^2 + y what are all the positive integers for x,y, z that satisfy this equation?
  45. anemone

    MHB Find the number of integers

    Find the number of integers $k$ in the set ${0,\,1,\,\cdots,\,2012}$ such that the combination number $\displaystyle {2012\choose k}=\dfrac{2012!}{k!(2012-k)!}$ is a multiple of 2012.
  46. J

    Python Retrieve integers from a file

    Hey. Im pretty new to Python (and programming in general in fact). I have received two different files, each containing 10 000 integers mixed up with some commas, colons, and \n, and in one of the files there are also negative numbers. I have tried all day retrieving those numbers using all...
  47. e2m2a

    Consecutive integers and relatively prime numbers

    Summary:: Interested in the history of the proof. Consecutive integer numbers are always relatively prime to each other. Does anyone know when this was proved? Was this known since Euclid's time or was this proved in modern times?
  48. ?

    I Are there any two pairs of integers with the same result in a specific function?

    Hello all, This is a problem of a different flavour from my usual shenanigans. I'm looking at a function $$f(m,n)=\frac{m^2n^2}{(m+n)(m-n)}$$ and am trying to determine if there are any two pairs of values ##(m_1,n_1)## and ##(m_2,n_2)## which evaluate to the same result. Assume that...
  49. M

    B Find triangles with areas that are integers

    It is pretty obvious that all right-angled triangles whose sides are integers will have areas which are also integers. Since either the base or height will be an even number, half base x height will always come out exactly. However, I have only found one non-right-angled triangle where this is...
  50. anemone

    MHB Expressing First 1000 Positive Integers as Floor Functions

    How many of the first 1000 positive integers can be expressed in the form $\lfloor 2x \rfloor+\lfloor 4x \rfloor+\lfloor 6x \rfloor+\lfloor 8x \rfloor$, where $x$ is a real number?
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