Integers Definition and 467 Threads

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...

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.

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...

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...

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...

Hello, I was wondering about the following combinatorial problem: given a natural number n, for example 20, in how many ways can I write it as the sum of positive integers? For example: 20 = 20 20 = 19 + 1 20 = 8 + 4 + 4 + 2 + 1 + 1 etc Please note that I ignore the order of the numbers, i.e...

Homework Statement The question is : Converting to short int, calculate 30064 + 30064 as short integers. Convert the answer to decimal ( the answer will be negative ). Homework Equations None. The Attempt at a Solution I converted 30064 to hex getting 7570. I then added...

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 =...

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...

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...

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...

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?

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...

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:

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.)

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?

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...

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

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...

Homework Statement Count the solutions in nonnegative integers x1,...,xk to x1 + ... + xk=n Homework Equations There's a Theorem in the chapter (shows that the answer is "n+k-1 choose k-1" but we're not allowed to use it. The Attempt at a Solution Well, obviously you can just...

I will post my solutions to a few problems I've considered here, I just need feedback from people who can tell me if my ideas are correct because I'm feeling shaky about the methods. Z7 is the dvr in the completion of the rationals w.r.t. the 7adic metric. Then for what integers a is there a...

given any two numbers a,b and an upper and lower bound for the sum of reciprocals of a certain class of integers between a and b, without any direct calculation how can optimal upper and lower bounds for the number of terms in the sum be found

given any two numbers k,j what is the largest sequence of integers such that the sum of any k consecutive terms is negative and the sum of any j consecutive terms is positive and how may we find a subset containing k of the first n numbers such that out of all subsets with k elements, this...

How would I make a new data type that would hold larger numbers than ones that we can currently use? for example I need a data type that can handle calculations from large factorials e.g 200!, = 200*199*198*197 ...... it's a really large number and can't be saved in ints/doubles/longs...

sorry for the many threads Let S_n denote the number of ways of expressing n as positive integrs.. S_1=1 , s_2=2, s_3=4 .. Prove that S_n=S_{n-1}+S_{n-2} ---S_1+1 no idea to prove that :

Hi Everyone! I decided recently to start reading a book that acts as a transition to upper level mathematics. The last section of the chapter introduces you to the different proof techniques and mathematical facts to produce mathematical proofs. I think I understand everything, but I wanted to...

Homework Statement Two n-digit integers (leading zeros allowed) are considered equivalent if one is a rearrangement of the other. (For example, 12033, 20331, and 01332 are considered equivalent five-digit integers.) If the digits 1, 3, and 7 can appear at most once, how many nonequivalent...

if Xn is a sequence of integers and Xn--->x as n----> infinity and x is an element of the reals. show that x must be an integer. i know that since the sequence is convergent it will be bounded. i don't however see how i can prove the above. thank you very much for any help.

For the theorem that states that in quadratic field Q[sqrt d], if d is congruent to 1 mod 4, then it is in the form (a + b sqrt d)/2 and if it's not, it's in the form a + b sqrt d where a and b are rational integers, is it saying that if a and b are rational integers and the quadratic number are...

does MATLAB supports long integers??

Find the number of positive integers in the range of 1976 through 3776 that are not divisible by 17. A={\mathbb{Z}^+\ \mbox{not divisible by 17}} Then I am looking for A^c I am not sure how to do this though.

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}...

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...

Alternative to "God Created the Integers" ~ Stephen Hawking Hi! I was thinking about buying a decent book covering the history of mathematics (from a fairly technical point of view), and "God Created the Integers" by Stephen Hawking seemed to be the perfect book. However, after having read some...

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...

hey, I've hit a bump in the road here. I'm trying to build code for a hash table out of my textbooks pseudo code, and I'm not quite sure how I'd represent a string of characters such as "thisISastring", as an integer for my key variable. I'm using division method if that helps at all. I know the...

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!

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.

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

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...

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...

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

Find integers s and t such that 1 = 7*s + 11*t. Show that s and t are not unique. I can find numbers that satisfy this question, t=2, s=-3 and t=-5, s=8, that show s and t are not unique, but this doesn't seem to be rigorous and I'm not sure where to start with proving this.

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 =...

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...

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...

\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

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...

Can someone guide me on how to express (.a1a2a3a4a5) base n as a quotient of base n integers. There is a bar over a3a4a5.

Homework Statement Show that the groups \textbf{Q} and \textbf{R} are not isomorphic (both under addition). This was already answered before https://www.physicsforums.com/showthread.php?t=294687", but using different theory (generators and cyclic groups). We haven't covered that stuff in class...