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. C

    Proof about relatively prime integers.

    This is not homework. If n is a positive odd integer then n and n+2^k are relatively prime. k is a positive integer. Let's assume for contradiction that n and n+2^k have a common factor. then it should divide their difference but their difference is 2^k and since n is odd it has...
  2. J

    MHB What values of k make x^2 + 12x + k factorable over the integers?

    try to determine all the positive values of k for which x^2 + 12x + k is factorable over the integers.
  3. P

    MHB Proving LCM Inequality for Positive Integers

    For all positive integers m > n, prove that : \operatorname{lcm}(m,n)+\operatorname{lcm}(m+1,n+1)>\frac{2mn}{\sqrt{m-n}}
  4. A

    MHB Find the finite sum of the square and cube exponent of integers

    Hey, it is clear for me that \sum_{i=1}^{n} i = \frac{n(n+1)}{2} how to find a formula for \sum_{i=1}^{n} i^2 \sum_{i=1}^{n} i^3 Thanks
  5. G

    Euler sum of positive integers = -1/12

    My question arises in the context of bosonic string theory … calculating the number of dimensions, consistent with Lorentz invariance, one finds a factor that is an infinite sum of mode numbers, i.e. positive integers … but it really goes back to Euler, and his argument that the sum of all...
  6. ssamsymn

    Is there a map from real numbers to non integers?

    Can you help me to construct a 1-1 mapping from real numbers onto non-integers? thanks
  7. N

    How to interpret quotient rings of gaussian integers

    Homework Statement This is just a small part of a larger question and is quite simple really. It's just that I want to confirm my understanding before moving on. What are some of the elements of Z[i]/I where I is an ideal generated by a non-zero non-unit integer. For the sake of argument...
  8. C

    Proving 2n Representable When n is: Converse True?

    Homework Statement Prove that 2n is representable when n is. Is the converse true? Representable is when a positive integer can be written as the sum of 2 integral squares. The Attempt at a Solution so n can be written as x^2+y^2 x and y are positive integers so then...
  9. P

    Find integers A and B such that A^2 +B^2 = 8585

    Homework Statement Find integers A and B such that A2 +B2 = 8585 Homework Equations The Attempt at a Solution So in this case, I already know the answer: Sum of 2 squares: 8585 = 67^2 + 64^2 = 76^2 + 53^2 = 88^2 + 29^2 = 92^2 + 11^2. I started off looking at the graph of the...
  10. anemone

    MHB Find positive integers for both a and b

    I have a question relating to solving for both a and b in the following question: Find positive integers a and b such that: $\displaystyle \left(\sqrt[3]{a}+\sqrt[3]{b}-1 \right)^2=49+20\sqrt[3]{6}$ This one appears to be tough because it doesn't seem right to expand the left hand side and...
  11. T

    Showing the Fundamental Group of S^1 is isomorphic to the integers

    Hi, I am reading J.P. May's book on "A Concise Course in Algebraic Topology" and have approached the calculation where \pi_{1}(S^{1})\congZ He defines a loop f_{n} by e^{2\pi ins} I want to show that [f_{n}][f_{m}]=[f_{m+n}] I understand this as trying to find a homotopy between...
  12. N

    Partition of Integers with mod

    Homework Statement Are the following subsets partitions of the set of integers? The set of integers divisible by 4, the set of integers equivalent to 1 mod 4, 2 mod 4, and 3 mod 4. Homework Equations The Attempt at a Solution Yes, it is a partition of the set of integers...
  13. T

    A mapping from an integral domain to non-negative integers, Abstract Algebra

    So just had this question as extra credit on a final: Let D be an integral domain, and suppose f is a non-constant map from D to the non-negative integers, with f(xy) = f(x)f(y). Show that if a has an inverse in D, f(a) = 1. Couldn't figure it out in time. I was thinking the way to go...
  14. N

    GCD & LCM of 6 Integers: Exponential Form

    Homework Statement What is the greatest common divisor and least common multiple of the integers below (answer should be left in exponential form)? 2^{3}, 3^{3}, 5^{1}, 11^{2}, 13^{3} and 2^{1}3^{3}5^{2}7^{4}13^{1}Homework Equations The Attempt at a Solution This is exactly the way the...
  15. T

    Logarithm question, finding all possible pairs of integers

    Homework Statement Find all possible pairs of integers a and n such that: log(1/n)(√(a+√(15)) - √(a -√(15)))=-1/2 (that's log to the base (1/n)) The Attempt at a Solution (1/n)^-1/2 = (√(a+√(15)) - √(a -√(15)) ∴ n^4 = (a+√(15) - (a -√(15) - 2√((a+√15)(a -√(15)) ∴ n^4 = =2√(15)...
  16. M

    Showing a function defined on the integers is continuous

    Homework Statement Suppose that the function f is defined only on the integers. Explain why it is continuous. Homework Equations The ε/δ definition of continuity at a point c: for all ε > 0, there exists a δ > 0 such that |f(x) - f(c)| ≤ ε whenever |x - c| ≤ δ The Attempt at a...
  17. B

    A lemma in the integers from calculus

    Suppose that M and N are natural numbers, such that N>M-1. Prove that N≥M The problem above is a rather minor lemma that I obtained while proving the ratio test from calculus. I was able to successfully prove the ratio test itself, but I took this lemma for granted, which I am now trying to...
  18. Square1

    Prove the set of integers is a commutative ring with identity

    How should one prove that the integers form a commutative ring? I am not sure exactly where to go with this and how much should be explicitly shown. A ring is meant to be a system that shares properties of Z and Zn. A commutative ring is a ring, with the commutative multiplication property...
  19. S

    Find positive integers a,b,n that satisfy this expression

    Homework Statement Are there any positive integers n, a and b such that 96n+88=a^2+b^2 Homework Equations The Attempt at a Solution It resembles the Pythagorean theorem but I'm not sure how that would help me solve it. I factored the LHS 2^3((2^2)(3)n+11)=a^2+b^2 How do I...
  20. B

    Proving Real Number x Exists Between Integers a & b

    The problem is "For every real number x, there exists integers a and b such that a≤x≤b and b-a=1" I am stuck on the first part of the proof. So in my proof I let a=x and b=x+1. Then x+1-x=1=b-a. But what I don't get why is that is safe for me assume here that a and b are not always...
  21. A

    Solving the Conundrum of Three Positive Integers

    1. Three consecutive positive integers are such that the sum of the squares of the first two and the product of the other two is 46. Find the numbers. Variables: x. Three numbers: (x), (x + 1), (x + 2) 2. (I think, although I'm not sure.) x2 + (x + 1)2 + (x + 1)(x + 2) = 46 3. x2...
  22. M

    For what positive integers n does 15|M

    Homework Statement For what positive integers n does 15|2^{2n}-1 Homework Equations We know 2^{2n}\equiv1mod15 I was thinking this might be helpful but not sure x^{2} ≡ −1 (mod p) is solvable if and only if p ≡ 1 (mod 4) The Attempt at a Solution I think that the answer is...
  23. Y

    Counting Quadruples of Integers: A Combinatorial Problem

    This should be a simple combinatorial problem. Suppose I have a number n which is a positive integer. Suppose, that there are four numbers a,b,c,d such that 0<=a<=b<=c<=d<=n. The question is how many quadruples of the form (a,b,c,d) can be formed out such arrangement? I realize that this is...
  24. W

    Consecutive integers such that the prime divisors of each is less or equal to 3

    For each integer n > 1, let p(n) denote the largest prime factor of n. Determine all triples (x; y; z) of distinct positive integers satisfying  x; y; z are in arithmetic progression,  p(xyz) <= 3. So far I have come up with 22k + 1, 22k + 1 + 22k, and 22k + 2 other than the solutions...
  25. A

    Consecutive integers divisible by a set of Primes

    I am having more than a little fun with this sequence of numbers and am looking for a better algorithm to find the next numbers in the sequence. Let Z be the set of the first n odd primes. Find two integers j and k that are relatively prime to all members of Z where every integer between...
  26. P

    Help proving subsets of the integers

    I just started taking a foundations of math course that deals with proofs and all that good stuff and I need help on a problem that I'm stuck on: Prove: Z={3k:k\inZ}\cup{3k+1:k\inZ}\cup{3k+2:k\inZ} Z in this problem is the set of integers This is all that's given. I thought maybe I...
  27. O

    MHB Every Number is between Two Consecutuve Integers

    Hello everyone, I want to prove that every number is between two consecutive integers. $x\in R$. The archimedean property furnishes a positive integer $m_1$ s.t. $m_1.1>x$. Apply the property again to get another positive integer $-m_2$ s.t. $-m_2.1>-x$. Now, we have $-m_2<x<m_1$. I stopped...
  28. S

    LaTeX Using the Symbol \Z for Integers in LaTex

    What's the best way to use Z as a symbol for the integers on the forum's LaTex? One source on the web (http://www.proofwiki.org/wiki/Symbols:Z) says the symbols for the integers can be written in LaTex as backslash Z. On the forum, that currently shows up as the two characters. \Z...
  29. P

    Non-Bijective Function from Integers to Integers

    Homework Statement Is it possible to find a non-bijective function from the integers to the integers such that: f(j+n)=f(j)+n where n is a fixed integer greater than or equal to 1 and j arbitrary integer. Homework Equations The Attempt at a Solution
  30. S

    Cardinality of 1-1 mappings of integers to themselves

    I think the cardinality of the set M of all 1-1 mappings of the integers to themselves should be the same as the cardinality of the real numbers, which I'll denote by \aleph_1 . My naive reasoning is: The cardinality of all subsets of the integers is \aleph_1 . A subset of the integers...
  31. S

    MHB Calculating $f_n(\theta)$ for Positive Integers $n$

    For a positive integer $n$, let $$f_n(\theta)=\tan \frac{\theta}{2}(1+\sec \theta)(1+\sec 2\theta)(1+\sec 4 \theta)\cdots (1+\sec2^n \theta)$$ Find the value of (i) $f_2 \left(\dfrac{\pi}{16} \right)$ (ii) $f_3 \left(\dfrac{\pi}{32} \right)$ (iii) $f_4 \left(\dfrac{\pi}{64} \right)$ (iv)...
  32. R

    Asymmetric Random Walk on the Set of Integers

    Homework Statement Give the value of u_0.Homework Equations Let p>q>0 with p+q = 1 and a = q/p < 1. Let X_n denote the random walk with transitions X_{n+1} = CASE 1: X_n + 1 with probability p and CASE 2: X_n - 1 with probability q. For i ≥ 0, we set u_i = P(X_n = 0 for some n ≥ 0|X_0 = i)...
  33. ?

    Sum of all possible products formed from a set of consecutive integers

    Homework Statement Say I have a set of consecutive integers up to N, for example up to 5, the set 1,2,3,4,5. I need to know a general formula for the value of the sum of all products that can be formed from a certain number of elements taken from such a set. For example with the 5 set, say I...
  34. R

    Gaussian integers, ring homomorphism and kernel

    Homework Statement let \varphi:\mathbb{Z}[i]\rightarrow \mathbb{Z}_{2} be the map for which \varphi(a+bi)=[a+b]_{2} a)verify that \varphi is a ring homomorphism and determine its kernel b) find a Gaussian integer z=a+bi s.t ker\varphi=(a+bi) c)show that ker\varphi is maximal ideal in...
  35. A

    Prove by Contradiction: For all integers x greater than 11

    Homework Statement Prove by Contradiction: For all integers x greater than 11, x equals the sum of two composite numbers. Homework Equations A composite number is any number that isn't prime To prove by contradiction implies that if you use a statement's as a negation, a contradiction...
  36. R

    The greatest common divisor of n integers

    Homework Statement define the gcd of a set of n integers, S={a_1...a_n} Prove that exists and can be written as q_1 a_1+...+q_n a_n for some integers, q_1...q_n Homework Equations Euclid's Algorithm? The Attempt at a Solution I have the statement that gcd(a_1...a_n) = min( gcd(a_i, a_j)...
  37. W

    How to minimise a function of integers?

    I want to find the positive integer, x, which minimises the following function:f(x) = (mn - 2(n-1)x - 1)^2where m and n are positive integers. I also have the further constraint that:\frac{m}{x} = \mathrm{positive \ integer}I guess calculus might not be a good route to take, since x can only...
  38. A

    Proof: The product of any 4 consecutive integers

    Homework Statement The product of any 4 consecutive integers will be one less than a perfect square. Homework Equations Well, a perfect square is a number that can be broken down to n*n where n is an integer. If a number is consecutive to another number that means it is exactly one more...
  39. C

    MHB Determine all solutions in positive integers a, b, and c to this equation.

    Determine all solutions for \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \ = \ 5, where \ \ a, \ b, \ and \ \ c \ \ are \ \ positive \ \ integers, \ \ and \ \ a <b < c.
  40. J

    The GCD forms a subgroup of the integers

    Let r and s be positive integers. Show that {nr + ms | n,m ε Z} is a subgroup of Z Proof: ---- "SKETCH" ----- Let r , s be positive integers. Consider the set {nr + ms | n,m ε Z}. We wish to show that this set is a subgroup of Z. Closure Let a , b ε {nr + ms | n,m ε...
  41. W

    Number theory find two smallest integers with same remainders

    Homework Statement Find the two smallest positive integers(different) having the remainders 2,3, and 2 when divided by 3,5, and 7 respectively. Homework Equations The Attempt at a Solution I got 23 and 128 as my answer. I tried using number theory where I started with 7x +2 as...
  42. V

    Only homomorphism from rationals to integers

    Kindly see the attached image.i can't understand the step where he write f(x)=f(1+1+...+1)=xf(1).But the homomorphism is from <Q,.> to <Z,+>
  43. M

    Euler's derivation of Riemann Zeta Function for even integers

    So Euler derived the analytic expression for the even integers of the Riemann Zeta Function. I was wondering if there is a link to his derivation somewhere? Also, is there anyone else who used a different method to get the same answer as Euler? Thank you
  44. L

    How can I generate random integers without bias using programming?

    In my book, it says the way to produce a random integer from, for example, 1-50 is to use srand() % 50 + 1. But wouldn't that give "1" the chance of showing up more often than other numbers? If srand is 0, then the random result is 1. If srand is 50, then the random result is also 1. The other...
  45. G

    Inifinte integers between each interval of time

    Hey guys! I am new here, and would like to ask a question that has been on my mind for a very long time. I've searched on the internet to find a solution to this question, but have come up with nothing, so I searched for a physics forum which could possibly put my question to rest. Here it is...
  46. K

    Integers, rationals and divisibility

    1.To prove - For any natural number n, the number N is not divisible by 3 2. N = n2+1 3. Dividing naturals into three classes according to remainder outcomes during division by 3 ie. 0,1,2 ; for any whole number k ---> 3k, 3k+1, 3k+2 And then substitute the values respectively to...
  47. N

    2^k+1 never divisible by u(8x-1) in integers

    It seems that there exists no integer k such that 2^k+1 is divisible by a positive integer n, if and only if n is of the form u(8x-1) (where u and x are also both positive integers). How could this be proved/disproved?
  48. W

    MHB Find "n" Given p=s and m=4,541,160 | Integer Solution

    p = product of 2 consecutive integers n-1 and n. s = sum of m consecutive integers, the first being n+1. s = p Example (n = 12, m = 8): p = 11 * 12 = 132 s = 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 132 If m = 4,541,160 then what's n ?
  49. S

    Equation with two variables (integers)

    Homework Statement Solve the equation (x & y are integers): (x^3+4)(xy^2-x^2y+3y^2-12)=x^6 Homework Equations The Attempt at a Solution xy^2-x^2y+3y^2-12=\frac{x^6}{x^3+4} \\ xy^2-x^2y+3y^2-12=x^3-4 + \frac{16}{x^3+4} \\ 16 \geq x^3+4 \\ x^3 \leq 12 That's all I can...
  50. N

    A≡b mod n true in ring of algebraic integers => true in ring of integers

    "a≡b mod n" true in ring of algebraic integers => true in ring of integers Hello, So I'm learning about number theory and somewhere it says that if a\equiv b \mod n is true in \Omega, being the ring of the algebraic integers, then the modular equivalence (is that the right terminology?) it...
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