Integers Definition and 474 Threads

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 all non-negative integers (x,y)

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

    How many integers to pick such that 2 of them have digit in common

    Homework Statement How many integers from 100 through 999 must you pick in order to be sure that at least two of them have a digit in common? (they don't have to be in the same place value) The Attempt at a Solution worst case scenario involves picking integers such that...
  4. D

    Cyclic Group Generators <z10, +> Mod 10 group of additive integers

    So I take <z10, +> this to be the group Z10 = {0,1,2,3,4,5,6,7,8,9} Mod 10 group of additive integers and I worked out the group generators, I won't do all of them but here's an example : <3> gives {3,6,9,2,5,8,1,4,7,0} on the other hand <2> gives {2,4,6,8,0} and that's it! but...
  5. Math Amateur

    MHB Algebraic Integers - Rotman - Proposition 2.70 - pages 118 - 119

    I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition). I am currently focussed on Proposition 2.70 [pages 118 - 119] concerning algebraic integers. I need help to the proof of part (iii) this Proposition. Proposition 2.70 and its proof...
  6. C

    Proof about product of 4 integers

    Homework Statement Prove that the product of four consecutive integers is always one less than a perfect square. The Attempt at a Solution I tried looking at the product (n-1)(n)(n+1)(n+2)=x^2-1 but i couldn't seem to get anything useful out of it. I added one to both sides . I tried...
  7. PsychonautQQ

    Show that a^2~0 or a^2~1 in Mod 4

    Homework Statement If a I san integer, show that a^2~0 or a^2~1 in mod 4 (~ represent equivalence) Homework Equations The Attempt at a Solution my ATTEMPT: I started with the division algorithm.. a = 2q + 1 for all odd numbers a = 2q + 0 for all even numbers then I squared the...
  8. evinda

    MHB Why aren't there coprime integers?

    Hey again! (Smile) I am looking at the following exercise: Why aren't there coprime integers $a,b>1$, such that $a^2 b^3=8100$? That's what I have tried: $b>1$,so it has a prime divisor $p$. $p$ can be $2 , 3 \text{ or } 5$. $p=2$: $$b=2k, k \in \mathbb{Z}$$ Then, $a^2 \cdot 2^3 \cdot...
  9. evinda

    MHB How many positive integers are not divisible by 2, 3, or 5 up to 120?

    Hi! (Smile) I am looking at this exercise: How many positive integers,that are not greater that $120$, do not get divided by $2,3 \text{ and } 5$? I thought to write $120$ as a product of prime numbers ($120=2^3 \cdot 3 \cdot 5^2$),and then find the number of multiples of $2,3,5$ and subtract...
  10. anorlunda

    Fortran [Fortran] Naming conventions for integers

    Long ago I learned Programming in FORTRAN. I got used to the convention that names starting with I,J,K,L,M,N were INTEGER while all other letters were REAL. I thought it was a convention of FORTRAN only. Since then, I came to realize that the same convention is widely used in science and math...
  11. matqkks

    Can Factorization of Integers Motivate Students in a First Number Theory Course?

    Why is factorization of integers important on a first number theory course? Where is factorization used in real life? Are there examples which have a real impact? I am looking for examples which will motivate students.
  12. matqkks

    MHB How Does Integer Factorization Impact Modern Cryptography and Beyond?

    Why is factorization of integers important? What are the real life applications of factorization? Are there are examples which have a real impact.
  13. anemone

    MHB Find the sum of all positive integers a

    Find the sum of all positive integers $a$ such that $\sqrt{\sqrt{(a+500)^2-250000}-a}$ is an integer.
  14. anemone

    MHB Find all positive integers a and b

    Find all positive integers $a$ and $b$ such that $a(a+2)(a+8)=3^b$.
  15. C

    Proof about relatively prime integers.

    Homework Statement Prove that if you have n+1 integers less than or equal to 2n then at least 2 are relatively prime. The Attempt at a Solution the book say integers but I am pretty sure this will only work in the natural numbers. there are n even numbers between 0 and 2n okay and none of...
  16. anemone

    MHB Find all solutions in integers of (a, b)

    Find all solutions in integers of the equation $a^3+(a+1)^3+(a+2)^3+\cdots+(a+7)^3=b^3$
  17. evinda

    MHB What Integers Between -100 and 400 Satisfy Specific Modular Conditions?

    Hello! :) I am given the following exercise: Which integers of the interval: $[-100,400]$ have the identity: divided by $11$,the remainder is $2$ and divided by $13$,the remainder is $3$. It is like that: $[x]_{11}=[2]_{11} \Rightarrow x \equiv 2(\mod 11) \Rightarrow 11 \mid x-2 \Rightarrow...
  18. H

    Increasing counterclockwise number of integers

    Homework Statement In the image below we start with the integer 1 marked in yellow. We will fill the rest of the table in counterclockwise manner with integers to infinity. What will be the sum of the numbers right above and below the number 2008? Homework Equations - The Attempt...
  19. anemone

    MHB How can $5^{1985}-1$ be factored into three integers greater than $5^{100}$?

    Factor $5^{1985}-1$ into a product of three integers, each of which is greater than $5^{100}$.
  20. J

    Least Common Multiple of an arbitrary number of positive integers

    I need an algorithm for LCM(k1, k2, ..., kn). Here's what I was thinking: any number ki that divides evenly into another number kj, set ki = 1 return k1*k2*...*kn I'm having trouble implementing it, though. int LCM(int* numsPtr, int size) { // assume size > 1 and that array only...
  21. anemone

    MHB Is the Product of Two Integers Greater Than 10^2009? Let's Prove It!

    Prove that $\large 3^{4^5}+4^{5^6}$ is the product of two integers, each at least $\large 10^{2009}$.
  22. anemone

    MHB Determine all pairs of integers

    Determine all pairs of integers $(a, b)$ satisfying the equation $b(a+b)=a^3-7a^2+11a-3$.
  23. mente oscura

    MHB Solve in positive integers for a² = 9555² + c²

    Hello. A simple question.Solve in all positive integers, for: a^2=9555^2+c^2 Please, you show the way of solving it. Regards.
  24. anemone

    MHB Determine the positive integers

    Determine the positive integers $z>y>x$ for which $\dfrac{1}{x}-\dfrac{1}{xy}-\dfrac{1}{xyz}=\dfrac{19}{97}$
  25. anemone

    MHB What is the solution to this equation in positive integers?

    Solve in positive integers for $577(bcd+b+c)=520(abcd+ab+ac+cd+1)$
  26. anemone

    MHB Find the number of integers "k"

    Find the number of integers $k$ with $1 \le k \le 2012$ for which there exist non-negative integers $a, b, c$ satisfying the equation $a^2(a^2+2c)-b^2(b^2+2c)=k$.
  27. J

    How to do this using series. Rep as ratio of two integers

    Homework Statement Express the number as a ratio of integers. 10.1(35) = 10.135353535 the part in the left in () is where is is over lined to indicate it is repeating Homework Equations Geometric series The Attempt at a Solution 10.1(35) = 10.1 + .035 = (101)/ (1000) +...
  28. ArcanaNoir

    MHB Finding GCD in Gaussian Integers

    The problem asks to find a generator of the principal ideal <6+7i, 5+3i> in Z[i]. It is my understanding that such a generator must be a greatest common divisor of 6+7i and 5+3i. So, let's call this guy d, we should have d(a+bi)=6+7i and d(c+di)=5+3i. I'm not really sure how to find d. If I...
  29. C

    Number of Primes between two integers

    Is there a formula to calculate the EXACT number of primes between two integers? There are many very good ways of ESTIMATING the number but I have found very few that give the EXACT number, and those that do essentially require the knowledge of primes before hand (Legendre and Miessel.) While...
  30. K

    MHB Can Every Integer Be Decomposed into an Odd Integer and a Power of 2?

    Prove that every n E N can be written as a product of odd integer and a non-negative integer power of 2. For instance: 36 = 22 * 9
  31. D

    Shuffling cards (a list of integers)

    Hi, I found this problem along with the solution: "Given 
an 
array 
of 
distinct
 integers,
 give 
an 
algorithm
 to 
randomly 
reorder 
the integers
 so 
that 
each
 possible 
reordering 
is 
equally 
likely. 

In 
other 
words, 
given 
a deck
 of 
cards,
 how
 can 
you
 shuffle 
them...
  32. I

    Mystery of the Integers: Unraveling Prob90

    https://www.physics.harvard.edu/uploads/files/undergrad/probweek/prob90.pdf https://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol90.pdf This is the puzzle I am trying to understand. Does anybody has any idea how the table on the top of the second page i being deducted...
  33. K

    MHB Let m and n be two integers. Prove that:

    Let m and n be two integers. Prove that if m2 + n2 is divisible by 4, then both m and n are even numbers Hint: Prove contrapositiveAttempt: Proof by Contrapositive. Assume m, n are odd numbers, showing that m^2 + n^2 is not divisible by 4. let: m= 2a + 1 (a,b are integers) n=2b+1 m^2+n^2 =...
  34. skate_nerd

    MHB Proving that not every non-empty set of integers has a least element

    I have a problem asking to prove the following statement is false: "Every non-empty set of integers has a least element". This seems pretty intuitively false, and so I tried to sum that up in the following way: Suppose we have a subset \(A\) in the "universe" \(X\). Let \(A=\{-n: n\in{N}\}\), a...
  35. G

    Summation problem (first N positive integers)

    Homework Statement Homework Equations so i kno the formula for the for the sum of the first N positive integers when i = 1The Attempt at a Solution i kno the answer = n^2(n+1)/2 but could someone explain step by step how you reduce it to get the final answer? as if I'm in kindergarten...
  36. anemone

    MHB How Many Integers Satisfy This Inequality?

    How many integers satisfy the following relation? |||x+9|-18|-98| \le 82
  37. anemone

    MHB Find the number of pairs of consecutive integers

    Consider the set of integers ${1000,1001,1002,...1998,1999,2000}$. There are times when a pair of consecutive integers can be added without "carrying": $1213 + 1214$ requires no carrying, whereas $1217 + 1218$ does require carrying. For how many pairs of consecutive integers is no...
  38. G

    MHB Fractals in relatively prime integers

    Greetings, humans! (Tongueout) I'm from Ukraine. My English is very bad. So I will use a Google Translate. In 2002, I came up with an interesting piece. I was only 14 years old. I was thinking about fractals and chaos theory, and did not want to learn. Did not want to learn, and were forced to...
  39. C

    Fundamental Property of Integers

    So there is a proof that the sum of any two even numbers is an even number. 2k + 2l = 2(k +l) We have written the sum as 2 times an integer. Therefore the sum of any two even numbers is an even number. An essential part of this proof is that k + l is an integer. How do we know this? Is it an...
  40. anemone

    MHB Sequence of Positive Integers Challenge

    Consider the sequence of positive integers which satisfies a_n=a_{n-1}^2+a_{n-2}^2+a_{n-3}^2 for all $n \ge 3$. Prove that if $a_k=1997$, then $k \le 3$.
  41. anemone

    MHB Solutions to x^3+...+y^3=y^3 in Integers

    Find all solutions in integers of the equation x^3+(x+1)^3+(x+2)^3+\cdots+(x+7)^3=y^3
  42. K

    MHB What is the meaning of Z^2 in set notation?

    Hello all, I'm having a lot of trouble when it comes to set notation. For instance, what does (the set of all integers) Z^2 mean? What values are contained in this set?Sorry if I didn't use the MATH tags right.
  43. paulmdrdo1

    MHB Converting a repeating decimal to ratio of integers

    0.17777777777 convert into a ratio.
  44. QuarkCharmer

    What's going on in this proof about integers?

    Homework Statement Prove that for every integer n>=8, there exists nonnegative integers a and b, such that n =3a+5b Homework Equations The Attempt at a Solution I'm trying to understand the proof of this. It goes as follows: I am having a hard time figuring out what is going...
  45. S

    Formula for adding all positive integers in a given interrval?

    Homework Statement I meant for the title to be, Sum of all EVEN integers A formula to add all even integers between two given points. (i.e.) All integers from 6 to 2000 ? 6+8+10+12 .. + 2000 The Attempt at a Solution The reason I ask is because I derived such an equation that will work for any...
  46. C

    Integers as the sum of 3 integers.

    Homework Statement Prove that every integer >17 can be written as the sum of 3 integers >1 that are pairwise relatively prime. The Attempt at a Solution I already proved the case for even integers. Now I am just working on the case for odd integers. I know that it has to be the sum of 3...
  47. C

    How can integers be written as sums of relatively prime integers?

    Homework Statement Prove that every integer bigger than 6 can be written as a sum of 2 integers bigger than 1 which are relatively prime. The Attempt at a Solution Ill first look at the case where our number is odd. Let x be an odd integer. I will just add (x-2)+2=x since x is odd so is x-2...
  48. anemone

    MHB What Positive Integers Satisfy a Specific Condition Involving Square Roots?

    Find all positive integers n for which \sqrt{n+\sqrt{1996}} exceeds \sqrt{n-1} by an integer.
  49. 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.
  50. E

    Proof involving the sum of squared integers

    Homework Statement Theorem: the numbers in the set {99, 999, 9999, ... } cannot be written as two squared integers, but at least one can be expressed as the sum of 3 squared integers. Homework Equations Well there are a lot of examples but let's go with 32 + 32 + 92 = 99 We may...
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