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. S

    Find a recurrence relation from a sequence of integers

    If you are given a sequence of integers such as: An=xn+yn where x and y are integers. and n=0,1,2,3... how would one find the recurrence relation? I tried writing An+1 in terms of An but it doesn't come out neatly because it doesn't translate so well. And there are terms raised to the n+1...
  2. B

    Division of Integers: Show n33-n Divisible by 15

    Homework Statement Show that for every integer n the number n33 - n is divisible by 15 Homework Equations The Attempt at a Solution Not sure what to do. I was thinking it might have something to do with both numbers are divisable by 3 ie the power = 3 x 11 and the divisor...
  3. S

    Relatively Prime Quadratic Integers

    Hello everybody. I found this example online and I was looking for some clarification. Assume 32 = \alpha\beta for \alpha,\beta relatively prime quadratic integers in \mathbb{Q}[i]. It can be shown that \alpha = \epsilon \gamma^2 for some unit \epsilon and some quadratic \gamma in...
  4. D

    Another integers mod 4 question

    Use the first isomorphism theorem to show the following: Z/(4Z) is isomorphic to Z4. There are other ones to solve I'm just using this as an example so I can figure out the thinking behind it. I can prove it with multiplication tables, but in reference to the F.I.T. I'm not sure how to start.
  5. R

    Prove For all integers a, 9 does not divide a^2 +3

    Homework Statement Trying to prove that for all integers a, 9 does not divide a^2 +3 Homework Equations there exists no k such that a^2+3 = 9k The Attempt at a Solution Tried assuming not the case, so assumed a is an integer and 9 does not divide a^2 +3 to try to prove a...
  6. H

    Construct a sequence whose set of limit points is exactly the set of integers?

    Homework Statement "Construct a sequence whose set of limit points is exactly the set of integers?" The Attempt at a Solution I need a sequence that will have an infinite number of terms that arrive at each of the integers, right? And since the sequence is indexed by the natural numbers...
  7. K

    Understanding Inverses & Max* for Integers

    Homework Statement If I have a group defined on the integers, by a*b=ab, how do I know if an inverse exists? Also, define * on the integers by a*b=max{a,b} Homework Equations The Attempt at a Solution I got 1/a as an inverse, but I'm thinking it's not a group since we don't...
  8. N

    In how many ways can I write n as a sum of integers?

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  9. L

    Add Short Ints: 30064 + 30064 = -60128

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  10. J

    Algebra, ring question with even integers

    We have E the set of even integers with ordinary addition Define new multiplication * on E defined as a*b = ab/2 where on the right hand side of the equation is just normal multiplication. I am just a bit confused i am trying to show Associative multiplication meaning i have to show (a*b)*c =...
  11. S

    Show that for any two integers a, b , (a+b)^2 ≡ a^2 + b^2 (mod 2)

    Show that for any two integers a, b , (a+b)^2 ≡ a^2 + b^2 (mod 2) I have my solution below i wanted someone to help chekc if i have done anything wrong. Thank You for your help. The thing that is going on here is that 2x = 0 (mod 2) for any x. If x = ab, then 2ab = 0 (mod 2). We see...
  12. D

    Programming Digits to Integers

    Object: Write a program that read digits and composes them into integers. For example, when you read 123( in characters) the program should print: "123(in integers) is 1 hundred and 2 tens and 3 ones". My first attempt is first read the number as a string then convert its characters to...
  13. ╔(σ_σ)╝

    Prove that the least upper bound of a set of a set of integers is

    Problem Statement: Prove that the least upper bound of a set of integers is an integer. Attempt: Using well ordered principle this is very trivial. However, is there another way? ANY comments or ideas relating to the topic would be highly appreciated. It is assumed that the set...
  14. G

    Name of the set of negative integers

    I know that N (natural numbers) is the set of non-negative integers, 0, 1, 2, 3, 4...infinity, and that Z is the set of all integers, both positive and negative. But what is the name or representation of the set of negative integers?
  15. S

    Prove: Row of 1000 Integers Becomes Identical Over Time

    A row contains 1000 integers The second row is formed by writing under each integer, the number of times it occurs in the first row.The third row is now constructed by writing under each number in the 2nd row, the number of times it occurs in the 2nd row.This is process is continued Prove...
  16. S

    Summing Positive Integers with 2^r3^s

    Show that every positive integer is a sum of one or more numbers of the form 2^r3^s, where r and s are nonnegative integers and no summand divides another. not my doubt just found it interesting so posted here :smile:
  17. P

    Exhibit a bijection between N and the set of all odd integers greater than 13

    Homework Statement Exhibit a bijection between N and the set of all odd integers greater than 13 Homework Equations The Attempt at a Solution I didn't have a template for the problem solving. Please check if I did it in the right way? (The way and order a professor will like to see.)
  18. J

    Proving an identity to have solutions over all the integers

    Hello, I was looking at some math problems and one kind caught my attention. The idea was to prove that let's say 3x+2y=5 has infinitely many solutions over the integers. Can someone show me the procedure how a problem like this might be solved?
  19. P

    Exploring the Finite Set of Integers Described in English

    It's simple for you mathematicians, but I'm a physician, I don't know much about set theory or logic and such, so it's difficult for me. Let M be the set of all integers that can be described in English in, say, ten lines of text. For example, "fourteen" or "seventy minus eight" or...
  20. M

    Find all integers n for which this fraction is an integer

    Find all integers n for which the fraction n ^ 3 + 2010 / (n ^ 2 + 2010) is equal to integer. please I need help :( Thank you
  21. A

    Proof involving a closed set of integers

    Homework Statement proove is either true of false let A be a set of integer closed under subtraction. if x and y are element of A, then x-ny is in A for any n in Z. Homework Equations n/a The Attempt at a Solution is there any proof, without induction? i suspect its true because any...
  22. J

    Count the solutions in nonnegative integers

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  23. S

    Solutions to equations in the padic integers (rationals)

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  24. H

    Sum of reciprocal of integers

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  25. H

    Sequence of integers and arithmetic progressions

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  26. V

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  27. D

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  28. A

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  29. S

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  30. S

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  31. S

    Quadratic Integers: Understanding the Theorem and Proving 32 = ab in Q[sqrt -1]

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  32. R

    MATLAB MATLAB & Long Integers: Support?

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  33. D

    Number of positive integers that are not divisible by 17

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  34. N

    Sum of n elements of a finite set of integers, 1 through s

    The general problem I'm trying to solve is the probability of rolling a total t on n s-sided dice. A good chunk of the problem is easy enough, but where I run into difficulty is this: How many combinations of dice will yield a sum total of t? Because the number set is limited, {a \choose n-1}...
  35. K

    Prove the sum of squares of two odd integers can't be a perfect square

    Homework Statement x^2+y^2=z^2 Homework Equations The Attempt at a Solution assume to the contrary that two odd numbers squared can be perfect squares. Then, x=2j+1 y=2k+1 (2j+1)^2 +(2k+1)^2=z^2 4j^2 +4j+1+4k^2+4k+1 =4j^2+4k^2+4j+4k+2=z^2 =2[2(j^2+K^2+j+k)+1)]=2s the...
  36. F

    Alternative to "God Created the Integers" by Stephen Hawking

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  37. G

    Defining Real Numbers Between Two Integers

    If we were to take any two integers on a real number line and mark a point (a number) halfway between the two, do the same in the range between the halfway point and each of the two numbers, and continue the process, would we be able to define all real numbers between the two integers (including...
  38. U

    How can I represent a string as an integer in C for a hash table?

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  39. N

    Product of any two even integers is a multiple of 4

    I am having the hardest time proving that "The product of any two even integers is a multiple of 4." My proof seems to be going in circles! Any guidance would be amazing!
  40. G

    Irrational numbers in infinite list of integers

    Is it safe to assume that the absolute value of sin x is greater than zero for all positive integer values of x? I have no real experience in number theory, and I don't know if you can say that there are no irrational numbers in an infinite list of integers.
  41. C

    Find All Integers Such that phi(n)=12

    I am trying to find all of the integers such that phi(n)=12. Clearly n=13 is one, but how do I do it for composite numbers? -Thanks
  42. F

    Convergence of a sequence of integers

    Homework Statement Given a Cauchy sequence of integers, prove that the sequence is eventually constant. [b]2. Relevant Definitions and Theorems Definition of Cauchy sequences and convergence Monotone convergence Every convergent sequence is bounded Anything relevant to integers...
  43. S

    Proving Congruent Integers: Tips & Advice

    Homework Statement I need to prove the following but have no idea how to do so. Let a,b, k be integers with k positive. If a is congruent to b(mod n), then ak is congruent to bk (mod n). Homework Equations The hint given is that I can assume the following proposition is true and that...
  44. K

    Proof on the divisibility of integers

    Homework Statement Let a,b be integers where a doesn't =0. Prove that if a divides b, and b divides a, then a=b or a=-bThe Attempt at a Solution I started out with b=aj and a=bk, where j,k are integers. Don't quite know how to proceed
  45. B

    Uniqueness of integers question

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  46. D

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    This is not actually a homework problem. Rather, it is a problem from Courant and Robbins' What is Mathematics?, Chapter 8: "The Calculus", page 409-410. Homework Statement Prove that for any rational k =/= -1 the same limit formula, N → k+1, and therefore the result: ∫a to b xk dx =...
  47. N

    Questions about the basic properties of Integers

    I am starting Number Theory this semester. My professor hands out notes but there is no textbook for the class. So hopefully you guys can help me with these seemingly easy problems. Z = {...,-5,-4,-3,-2,-1,0,1,2,3,4,5,...} Z is used to denote the set of integers 1) Show that if a is an...
  48. N

    Product of two consecutive integers

    Homework Statement Prove that n^2+n is even. Where n is a positive integer. Homework Equations n^2+n The Attempt at a Solution n^2+n = n(n+1) One of which must be even, and therefore the product of 2 and an integer k. n = 2k, \left \left 2*(k(n+1)) or n+1 = 2k...
  49. B

    Why are the trivial zeros negative even integers?

    \varsigma(s) = \sum^{\infty}_{n=1}n^{-s} If you substitute a trivial zero, let's say -2. Wouldn't it be \varsigma(s) = \sum^{\infty}_{n=1} = 1^2 + 2^2 + 3^2 + 4^2 + . . . How would this series be equals to zero? Thanks
  50. X

    Find all solutions in positive integers

    Find all solutions in positive integers a; b; c to the equation a!b! = a! + b! + c! I have rearranged and got (a!-1)(b!-1) = c!+1 And the only solutions I can find are a=3 b=3 c=4 but I can't be sure that they are the only ones. How would I go about finding other solutions? I have...
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