Integers Definition and 467 Threads

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...

Homework Statement Show that for all integers n>2, n does not divide n^2+2. 2. The attempt at a solution I believe this solution can be solved by induction, I just don't know how to phrase it recursively. For all n>2, n^2+2 mod n ≠ 0 Base case n=3 3^2 + 2 =11 11 mod 3 = 2 ≠ 0...

Find all positive integers n such that $\phi(n)=6$. n>1 so we can write n as a product of primes, say $p_{1},...,p_{t}$ are the prime factors. Then, using the multiplicative property, we find that $n(1-p_{1})...(1-p_{t})=6p_{1}...p_{t}$. I've tried using odd/even arguments to deduce...

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...

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

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

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

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...

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

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...

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...

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...

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...

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...

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...

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...

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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)...

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)...

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...

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...

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)...

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...

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...

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.

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...

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...

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,+>

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

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...

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...

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...

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?

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 ?

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...

"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...