Integer Definition and 620 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. I

    Solving Hexadecimal Unsigned 8-bit Integer Multiplication

    Homework Statement a) 50 b) 23 Product of the hexadecimal unsigned 8-bit integers Homework Equations Step Action |||||||||| Multiplicand ||||||| Product/Multiplier 0 Initial Vals ||||||||| 101 000 ||||||| 000 000 010 011 The Attempt at a Solution How...
  2. B

    Division of integer square by 4 leaves remainder 0 or 1

    Hi, I am looking for an explanation, if any, on why every integer square leaves remainder 0 or 1 on division by 4. Appreciate your time and help bluemoon2188
  3. D

    Explaining discontinuity in a greatest integer function

    Homework Statement Find the numbers, if any, where the function is discontinuous. f(x) = [[x - 2]] The attempt at a solution function is discontinuous for all integer values of x. I know that this is the obvious answer, however I am required to explain this in clear mathematical...
  4. S

    What is the Integer Program Formulation for Maximizing Scores in a Word Puzzle?

    Homework Statement Consider the following puzzle. You are to choose 4 three-letter "words" from the following list: ECB EFH BEJ GGE HIJ CDE GEG CBJ For each word, you earn a score equal to the position that the word's third letter appears in the alphabet. For example, ECB earns a score...
  5. L

    Integer Spin and Half Spin: What's the Difference? (Bosons vs. Fermions)

    bosons have integer spin, fermions have half spin, what does that mean? why bosons (integer spin) is able to avoid pauli's exclusion principle?
  6. S

    Integer factorization given enough primes

    I realize that this might seems to be a strange question, but after doing some coding i realized the following. to brute force the factorization of all numbers less than one million takes around 665 million tests (i.e. does this number divide the original). to do it "smarter" (least i...
  7. G

    Generating division problems with integer results

    I'm not sure at all how best to approach this problem. Basically, the application is for practicing mental arithmetic. Most of the features are pretty easy, with the exception of division, which is the topic here. I accept a minimum and maximum for the first and last numbers in the division...
  8. O

    Assignment makes pointer from integer without cast

    "assignment makes pointer from integer without cast" elephant* get_elephants() { elephant *current, *first; int response; /* create first node */ first = (elephant*)calloc(1,sizeof(elephant)); /* THIS LINE */ current = first...
  9. F

    Finding the aproximate integer Sqrt of a large prime

    I am trying to understand how I can find the square root of a large prime number in the form of an integer value, the portion after the decimal is irelevant. The numbers I wish to compute range around 188 multiplied by 3 to the power of 6548 plus 1, as an example. so let's say in excess of...
  10. K

    Smallest Positive Integer N for which tow(n)=6 is True

    Find the form of all n in N satifying tow(n)=6. (sorry don't know how to write this Tex). What is the smallest positive integer n for which this is true? tow(n)=6 so 6=(1+a1)(1+a2)----(1+ak) where n=p1^a1p2^a2---pk^ak
  11. D

    Proof: Showing C(n,0) < C(n,1) <...<C(n, floor(n/2)) = C(n, ceiling(n/2))

    Show that if n is a positive integer, then C(n,0) < C(n,1) <...<C(n, floor(n/2)) = C Homework Statement Show that if n is a positive integer, then C(n,0) < C(n,1) <...<C(n, floor(n/2)) = C(n, ceiling(n/2)) > ... > C(n, n) Homework Equations none The Attempt at a Solution Seems pretty...
  12. R

    Why Does \( a^3 \equiv a \mod 3 \) Hold for All Integers \( a \)?

    Homework Statement If a \in Z, then a^3 \equiv a (mod 3) . 2. The attempt at a solution Proof: Suppose a \in Z. Thus a is either odd or even. Case 1: Let a be even. Thus a = 2k, for some k \in Z. So a^3 - a = 8k^3 -2k = 2(4k^3 - k) = 2(k)(2k - 1)(2k + 1). Notice that, for all k...
  13. N

    So, do negative prime numbers exist?

    I know that the fundamental theorem of arithmetic states that any integer greater than 1 can be written as an unique prime factorization. I was wondering if there is any concept of negative prime numbers, because any integer greater than 1 or less than -1 should be able to be written as n = p1...
  14. J

    C code, converting float into integer

    Hi guys, i am using the C language and have created the following function: void counter (double c) { FILE *fp; char output[]="output.xls"; int n,p=0; int r=0; int i,j; float x=0; fp=fopen(output,"w"); fprintf(fp,"rmax\tnumber of particles within rmax\r\r\n"); for(n=1;n<=10;n++)...
  15. A

    Sums of Integer Powers - C(s) Convergence

    Hello all, Is there a closed form expression for the convergence of C(s) = \sum_{n=1}^{N} n^{s} Cheers, Adrian
  16. D

    Number theory: confused about the phrase an integer of the form

    Number theory: confused about the phrase "an integer of the form" Homework Statement Prove that any prime of the form 3k+1 is of the form 6k+1 Homework Equations The Attempt at a Solution I'm not sure where to start at all. I tried rewriting 3k+1 as 6k+2=6k+(6-4)=6(k+1)-4. But...
  17. S

    Finding integer numbers using basic operations

    Hello everyone! I am trying to construct an algorithm for the following problem and was wondering if there is any existing body of knowledge on this. Please forgive me if this is inappropriate (or ridiculous) but I am totally foreign to number theory. It goes like this: You are given n...
  18. T

    Arbitrary-precision Integer Calculator for bignum arithmetic

    Beaconaut APICalc 2 just released on Jan.18, 2011, which is an arbitrary-precision integer calculator for bignum arithmetic, cryptography analysis and number theory research... http://www.beaconaut.com/forums/default.aspx?g=posts&t=13"
  19. V

    Solving Inequalities with Integers

    Homework Statement For what integer value of X is 3x+5> 11 and x-3 <1?
  20. S

    How to avoid a character value in integer or floating variables

    Hello everyone Thanks for viewing this topic for me... my question is when I tried to check user inputs a valid age it works only if a number is entered. If by mistake user inputs a character value (e.g. any character a to z or any symbol) this loop runs infinite How can I check if a...
  21. B

    Solving equations with greatest integer function

    Homework Statement I can't find a step by step explanation for solving these types of equations eg. 99 = [2x+1]/3 Homework Equations eg. 99 = [2x+1]/3 or 48 = 4[2x/3] How do you handle the multipliers iand constants inside the brackets? thx [b]3. The Attempt at a...
  22. T

    Solving the "n" Challenge - Find the Smallest Integer

    Homework Statement This is just a problem i came up with while doing in equations, and recently saw in a book too. Find the smallest integer n, for which n! > 10n The Attempt at a Solution The trial method (with help from the computer) yields 25 to be the answer. After a...
  23. O

    Can this equation be solved using integer numbers?

    How to solve that eqation? 3^x=4y+5
  24. A

    Diophantine equation in the integer quantum hall effect

    Hi everybody! I'm studying the IQHE and I want to understand the rise of the diophantine equation. I read the thouless article but it was no so conclusive. I've also read the kohmoto article (phys rev B 1989) and he says that that property comes from the darboux theorem but i don't...
  25. M

    Proving 12 - 22 + 32 - 42 + ... + (-1)n+1)n2 for Every Positive Integer n

    Homework Statement This is a proof problem in the mathematical induction section of my textbook. I am having trouble with question (c). Result: 12 + 22 + 32 + ... + n2 = n(n+1)(2n+1)/6 for every positive integer n. (a) Use the result to determine the formula for 22 + 42 + 62 + ... + (2n)2 for...
  26. N

    Odd Integer and Multiple of Four

    Homework Statement Suppose that n is an odd integer. Prove that n is either one greater than a multiple of 4 or one less than a multiple of 4. Homework Equations N/A The Attempt at a Solution I realize that this is going to be a direct proof. However, I am stumped on where to...
  27. N

    Proving the Integer Multiples of A for (4n+3) and (2n+1): Homework Statement

    Homework Statement For some integer n, a|(4n+3) and a|(2n+1). Therefore, 4n+3 is an integer multiple of a, as well as (2n+1). Prove or disprove that a=+/-1. Homework Equations N/A The Attempt at a Solution I have been working on this one for quite some time now, but I cannot...
  28. H

    Need a hint for calculating the integer part of

    Homework Statement Calculate the integer part ( whole number part ) of 1 + 1/sqrt ( 2 ) + 1 / sqrt ( 3 ) + 1 / sqrt ( 4 ) + 1 / sqrt ( 5 ) + ... + 1 / sqrt ( 1 000 000 ) Any hints? Thanks, I'd appreciate the help
  29. A

    Integer Inequality Homework: Proving Existence of Integer m

    Homework Statement let a,b be positive integer , c is real number, and -a<c<b i want to show there exist integer m, -a \leq m \leq b such that m-1 \leq c<m i don't know any easy method, but this is where i got now, Let set S=[m|-a \leq m \leq b] So by contradiction, suppose that for all m...
  30. J

    Frequency of Greatest Integer quotients

    When looking for potential factors of an integer, I noticed that I could predict the frequency of the greatest integer function quotient and use this prediction to jump to the next potential factor. See the attachment for an example. I don't know if someone else had already discovered this...
  31. W

    Proving Existence of Positive Integer Multiple with 0s & 1s

    Homework Statement Let k be any positive integer. Prove that there exists a positive integer multiple n of k such that the only digits in n are 0s and 1s. (Use the pigeonhole principle.) Homework Equations The General Pigeonhole Principle If more than mk things are distributed into k...
  32. V

    Prime values of integer polynomials

    Hey there, physics forums! A question occurred to me the other day: Is it true that if f \in \mathbb{Z}[x] is monic and irreducible over \mathbb{Q} , then for at least one a \in \mathbb{Z} , f(a) is prime? I can't prove it, but I suspect it's true. Does anyone know if this problem...
  33. C

    Simplifying √(3+2√(2)): Find the Integer and Irrational

    Homework Statement √(3+2√(2)) can be simplified into a fairly simple sum of two number: one is an integer and one is an irrational number . Find them and show you are correct by squaring both sides of the "equation" Homework Equations The Attempt at a Solution
  34. B

    Understanding UFD's with Quadratic Integer & Norm Questions

    Hello PhysicsForums! I was reading up on UFD's and I came up with a few quick questions. 1. Why don't the integers of Q[\sqrt{-5}] form a UFD? I was trying to tie in the quadratic integers that divide 6 to help me understand this, but I am stuck. 2. Why is Z a UFD? 3. Assuming Q[\sqrt{d}] is...
  35. M

    Proving the Limit of f(x)-g(x) for Even Integer Polynomials

    Homework Statement I want to prove that a polynomial f(x) and a polynomial g(x) with degrees of k,n where k,n are positive even integer, n>k that limit x-> - infinity of f(x)-g(x)=-infinity Homework Equations a polynomial can be written as a1x^n+a2x^(n-1)...+a(n-1)x+an The...
  36. C

    Deceptively Simple Integer Problem

    [bHomework Equations none The Attempt at a SolutionHomework Statement Homework Equations The Attempt at a Solution
  37. K

    Proving "[a]_n ∩ [b]_n is Empty or Equals [b]_n

    Homework Statement Prove that either [a]_{n}\cap[b]_{n}=empty set or [a]_{n}=[b]_{n}.Homework Equations The Attempt at a Solution I want to assume there is an element x in [a]_{n}\cap[b]_{n} and show this implies [a]_{n}=[b]_{n}. This tells me x is in [a]_{n} and [b]_{n}. That's where I get stuck.
  38. T

    Frobenius method indical roots differing by a postive integer

    Homework Statement For what values of the constant K does the differential equation y"-(1/4 +K/x)y=0 (0<X< infinity) have a nontrival solution vanishing at x=0 and x= infinity ? Homework Equations Hints that were suggested for my prof were: For large x...
  39. R

    Solving Observation 4.2: Showing Every Integer Has Exactly One Binary Expansion

    I have 2 questions here. I know there is a lot of text, but don;t be scared or deterred from helping please, it is simple to understand and mostly just text/explanation. The Second one is very long, the first not so much. 1)Problem: Complete the proof of Observation 4.2 by showing that every...
  40. A

    Solve explicitly integer solution

    Homework Statement find p,q,r,s integer solution \frac{1}{p^2}+\frac{1}{q^2}+\frac{1}{r^2}+\frac{1}{s^2}=1 Homework Equations here some alternate form you can see http://www.wolframalpha.com/input/?i=1/p^2%2B1/q^2%2B1/r^2%2B1/s^2%3D1 The Attempt at a Solution i don't know...
  41. A

    Solving Integer Equation: 1/x + 1/y = 1

    Homework Statement let x,y \in N. Solve the equation \frac{1}{x}+\frac{1}{y}=1 Homework Equations n/a The Attempt at a Solution so i can see x=y=2 is the solution, hmm, isn't there any other solution. and, this is one of three of my assignment question for 15% continuous...
  42. A

    Prove that if a and b are both odd integer

    Homework Statement prove that if a and b are both odd integer, then 16|(a^2+b^2-2) Homework Equations n/a The Attempt at a Solution let a=2m+1 \ and \ b=2n+1, then a^2+b^2-2=4(m(m+1)+n(n+1)) so its divisible by 4, and also divisible 8 since m(m+1) \ and \ n(n+1) are even. so i only prove...
  43. E

    Explaining Ultimate Factorial Value in x^n Integer Series

    In the situation where differences between consecutive squares, (or consecutive cubes, consecutive x^4, etc.) are calculated, then the differences between those differences are calculated, and then the differences of those differences, and so on until you reach a constant number at a deep...
  44. 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
  45. G

    Why does Chern class belong to INTEGER cohomology class?

    Hi all, I am new to topology & geometry. I skimmed through the subject in various books so I might be asking simple questions. My question is how can we say Chern's class belong to Integer rather than Real cohomology class? I assume one defines it through the characteristic polynomial of...
  46. A

    N2 has the form 3k or 3k+1 for some integer k

    Homework Statement n2 has the form 3k or 3k+1 for some integer k Homework Equations n/a The Attempt at a Solution i've tried to table it for me to see, but still i don't have the general idea, should i show n2 is either divisable by 3k or 3k+1? if yes i still don't know owho
  47. B

    C/C++ How can I implement this in C++ or VB.net?

    Please help - I am helping a student who is trying to translate Ong-Schnorr to C++ or VB.net language. My major was electronics so I am not sure how to interpret inverse random integer k. Any one ? or suggest me a reference to read on - much thanks, newbie Ong-Schnorr-Shamir (from Briuce's...
  48. B

    Proving Only 2 Positive Integer Solutions for x3 = y2 – 15

    A problem asks to find all possible pairs (x, y) of positive integers that satisfy the equation: x3 = y2 – 15 There are 2 pairs (so far) that satisfy the equation: x = 1, y = 4 x = 109, y = 1138 It's possible that these 2 points are the only two positive integer solutions...
  49. T

    Proof of Integer Parts of Real numbers

    I am struggling to understand the proof for integer parts of real numbers. I have used to mean less than or equal to because I could not work out how to type it in. I need to show that: ∃ unique n ∈ Z s.t. nx<n+1 The proof given is the following: Let A={k∈Z : kx} This is a...
  50. A

    Mathematica: How do I program this? Square free part of an integer

    I am attempting to program Mathematica to multiply the square free terms of an integer. Basically say we are looking at 252, its prime factors are 2^2*3^2*7. So what I want to do is have Mathematica return to me just 2*3*7 when I enter 252. So I have this S := FactorInteger[252]...
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