Integers Definition and 474 Threads

What is the sum of all the even integers between 1 and 101? Is there an easier way besides using the formula: (B-A+1)(B+A)/2? It just takes too much time.

$f(n)=\underbrace{111--1}\underbrace{222--2}$ $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n$ prove:$f(n)$ is a product of two consecutive positive integers for all $n\in N$

$a_1=3,a_2=5757,\,\, a_n=\dfrac {7(a_a+a_2+-------+a_n)}{n}, \,\, (n\geq2)\,\, prove \,\,each\,\, term\,\, of\,\, a_n\,\, is \,\, an \,\, integer$ correction : $a_1=3,a_2=5757,\,\, a_n=\dfrac {7(a_1+a_2+-------+a_{n-1})}{n}, \,\, (n\geq2)\,\, prove \,\,each\,\, term\,\, of\,\, a_n\,\, is \,\...

hi all so this is a very basic question i think and i feel very bad for tumbling here but still i need to clear this, so from childhood i was taught that the negative numbers are less than positive but now when i am studying limits and functions i came across absolute function and it said |-x| =...

I realize this question may not have an obvious answer, but I am curious: I am using Gaussian and Eisenstein integer domains for geometry research. The Gaussian integers can be described using pairs of rational integers (referring to the real and imaginary dimensions of the complex plane). And...

Find the pairs of nonnegative integers, $(m,n)$, which obey the equality: \[(m-n)^2(n^2-m) = 4m^2n\] So far, I haven´t found a single pair, but I cannot prove, that the set of solutions is empty. Perhaps, someone can help me to crack this nut?

I am a physics student trying to self-learn Chern numbers and Chern class. The book I am learning (Nakahara) introduces the total Chern class as an invariant polynomial of local curvature form ##F## ## P(F) = \det (I + t\frac{{iF}}{{2\pi }}) = \sum\limits_{r = 0}^k {{t^r}{P_r}(F)} ## and each...

Find all integers, $n$, such that the set $\{1,2,3,4, ...,n\}$ can be written as the disjoint union of the subsets, $A$, $B$ and $C$ -whose sums of elements are equal.

Let $a, b \le 2015$ be positive integers. What is the number of integers of the form: \[ \frac{a^4+b^4}{625}? \]

Homework Statement Let n be an integer. Prove that the integers 6n-1, 6n+1, 6n+2, 6n+3, and 6n+5 are pairwise relatively prime. Homework EquationsThe Attempt at a Solution I tried to prove that the first two integers in the list are relatively prime. (6n-1)-(6n+1)=1 (trying to eliminate...

Homework Statement Show that out of a set of ten consecutive integers, at least one is coprime to all of the others. Homework Equations Lemma: Out of a set of n consecutive integers, exactly one is divisible by n. (Given). The Attempt at a Solution Let a1, a2...a10 be consecutive integers...

Hi All I have the following question. I have reviewed my notes but have not been able to crack this. I tried two different ways, both wrong. First: (4-6*\sqrt2)^2= 16-24*\sqrt2-24*\sqrt2+(36*2) = 88-218*\sqrt2 so, $x=88$ and $y=218$My second method was (4-6*\sqrt2)^2= 4^2-(6*2)=28...

Homework Statement (22016+ 5)m + 22015 = 2n + 1[/B] find every n and m pairs as they are positive integersThe Attempt at a Solution (22016+ 5)0 + 22015 = 22015 + 1[/B] so one pair is m= 0 , n = 2015 if m =1 the equation is meaningless if m> 1 so there are really amount of powers that...

Is there a way to identify a factorial without referring to computation of a factorial? For example, is there a way to identify 5040 as a factorial and a way to identify 5050 as not a factorial?

This Stephen Hawking book gathers the most important books along human history. In theory, this book is just divulgative, but, Gäuss, Riemann, Gödel... books included in Hawking's are not meant to be divulgative. So, any of you who read the book(/books if separeted) think it's maybe too hard...

I have the following problem: What is the sum of the first 40 positive odd integers? I look at the solution, and it says that "The sum of the first 40 positive odd integers is ##1 + 3 + 5 + \dotsm + 77 + 79##. And then it goes on with the solution. My question is, how do I find that 79 is...

Integers ##p## and ##q## having no common factors implies ##p^2## and ##q^2## have no common factors. Could you prove this without using the fundamental theorem of arithmetic (every integer greater than 1 either is prime itself or is the product of prime numbers, and that this product is unique...

Homework Statement This problem is from MIT OpenCourseWare- a diagram is attached to clarify certain definitions. I'd like to check my answers. The degree sequence of a simple graph is the weakly decreasing sequence of degrees of its vertices. For example, the degree sequence for the 5-vertex...

Hello, Say I have some integer n in some interval such that, gcd(n, k) = gcd(n + 1, k) = 1, for some composite odd integer k >= 9 I want to know if such n exists in that interval. To know that one exists suffices. I have tried to think in terms of modular arithmetic where for all primes in k...

Has anyone else spotted an unusual set of three different integers A, B, & C such that A^n + B^n - C^n = A + B - C > 0 (n > 1 and A x B x C > 0) I leave the reader to see if they can find this set, or to ask me what they are.

Hi, I’m having a problem with a particular case of binomial proportion. I want calculate a confidence Intervals for a binomial proportion for an efficiency. This kind of intervals are usually defined for ratios between integers numbers but in my case I had to subtract from both numerators and...

When I pick a random number on a number line made out of integers, starting from zero and expanding infinite to the right, what can I say about the position of this random number ? To the right the amount of numbers is infinite. To the left is an amount, a number, so that is finite, but it has...

Iam working through Spivak calculus now. The book defines natural numbers as of form N=1,2,3,4... Iam able to prove that every natural number is either odd or even. How can I extend to Z, integers? In one of the problems, Spivak says we can write any integer of the form 3n, 3n+1, 3n+2.( n is...

For all positive integers $n$, $r$, and $s$, if $rs \le n$ then $r \le\sqrt{n}$ or $s \le \sqrt{n}$ Proof: Suppose $r$ , $s$ and $n$, are integers and $r > \sqrt{n}$ and $ s > \sqrt{e}$. Multiply both sides of the first inequality by $s$. I get $sr > s\sqrt{n} $, but the book gives $rs >...

For all integers $a$, $b$, and $c$, if $a \nmid bc$, then $a \nmid b$ I need to prove this by contraposition. I get that by definition, $b = ak$ for some integer $k$. But I don't get the following step in the textbook: $bc = (ak)c = a(kc)$ I'm guessing there is something very obvious I'm...

Homework Statement Hello all I apologize for the triviality of this: Im new to this stuff (its easy but unfamiliar) I was wondering if someone could verify this: Find the G.C.D of a= 14+2i and b=21+26i . a,b \in \mathbb{Z} [ i ] - Gaussian Integers Homework Equations None The Attempt...

Homework Statement [/B] Show that the statement holds for all positive integers n 2n ≤ 2^n Homework Equations Axiom of induction: 1 ∈ S and k ∈ S ⇒ k + 1 ∈ S The Attempt at a Solution Let S be set of integers 2(1) ≤ 2^1, so S contains 1 k ∈ S, 2k ≤ 2^k I want to show k + 1 ∈ S, 2k +...

I'm doing the exercises from Introduction to Analytic Number Theory by A.J. Hildebrand (online pdf lecture notes) from Chapter 2: Arithmetic Functions II - Asymptotic Estimates, and some of them leave me stumped... 1. Homework Statement Problem 2.14: Obtain an asymptotic estimate with error...

I am studying this in the context of group/ring theory, ideals etc. So I post it here and not in the number theory section. 6. Suppose gcd(a,b)=1 and c|ab. Prove That there exist integers r and s such that c=rs, r|a, s|b and gcd (r,s)=1. One of my attempts: From gcd(a,b)=1 there exist s',t'...

Solve for positive integers the equation $ab+bc+ca=1+5\sqrt{a^2+b^2+c^2}$.

Find all solutions in positive integers $z<y<x$ to the equation $x^3+y^3+z^3=(x+y+z)^2$.

Prove that the equation $a^4+b^4+c^4-2a^2b^2-2b^2c^2-2a^2c^2=24$ has no solution in integers $a,\,b,\,c$.

The integers $x$ and $y$ have the property that for every non-negative integer $n$, the number $2^nx+y$ is a perfect square. Show that $x=0$.

Homework Statement Let * and = be defined by a*b means a - b is an element of the integers and a = b means that a - b is an element of the rationals. Suppose there is a mapping P: (* equivalence classes over the real numbers) --> (= equivalence classes over the real numbers). show that this...

Homework Statement Show that if m and n are integers such that 4|m2+n2, then 4|mn Homework EquationsThe Attempt at a Solution Since 4 divides m2+n2, then we can say that m2+n2 = 4k, where k is an integer. I haven't done any mathematical proofs of any kind yet, but we were supposed to see if...

Hello! (Smirk) Proposition The set $\mathbb{Z}$ of integers is countable. Proof $\mathbb{Z}$ is an infinite set since $\{ +n: n \in \omega \} \subset \mathbb{Z}$. $$+n= [\langle n, 0 \rangle]=\{ \langle k,l \rangle: k+n=l\}$$ We define the function $f: \omega^2 \to \mathbb{Z}$ with...

Hi! (Smirk) According to my lecture notes: Constitution of integers Equivalence relation $R$ on $\omega \times \omega$ For $\langle m,n \rangle \in \omega^2$ and $\langle k,l \rangle \in \omega^2$ we say that $\langle m,n \rangle R \langle k,l \rangle$ iff $m+l=n+k$. First Step: $R$ is an...

Hello! (Smile) How could we show that we can sort $m$ integers with values between $0$ and $m^2-1$ in $O(m)$ time? (Thinking)

Hey! :o Let $G=(V,E)$ be a directed acyclic graph. I have to write an algorithm to assign integers to the vertices of $G$ such that if there is a directed edge from vertex $i$ to vertex $j$, then $i$ is less than $j$.Is it maybe as followed?? L ← Set of integers S ← Set of all...

Find all pairs of integers $x,\,y>3$ such that there exist infinitely many positive integers $k$ for which $\dfrac{k^x+k-1}{k^y+k^2-1}$ is an integer.

Find all integers $k$ for which $x^2-x+k$ divides $x^{13}+x+90$.

Prove the following theorem: Theorem For a prime number p and integer i, if 0 < i < p then p!/[(p− i)! * i] * 1/p Not sure how to go about this. I wanted to do a direct proof and this is what I've got so far. let i = p-n then p!/[(p-n)!*(p-n)] but that doesn't exactly prove much.

In Chapter 1: "Integral Domains", of Saban Alaca and Kenneth S. Williams' (A&W) book "Introductory Algebraic Number Theory", the set of all Eisenstein integers, \mathbb{Z} + \mathbb{Z} \omega is defined as follows:https://www.physicsforums.com/attachments/3392Then, Exercise 2 on page 23 of A&W...

Homework Statement Prove that if x,y, and z are integers and xyz=1, then x=y=z=1 or two equal -1 and the other is 1. 2. Homework Equations The Attempt at a Solution Clearly, if I plug in 1 for each variable, or -1 in for two variables and 1 for the remaining variable, then the equation is...

In John Stillwell's book: Elements of Number Theory, Chapter 6 concerns the Gaussian integers, \mathbb{Z} = \{ a + bi \ | \ a, b \in \mathbb{Z} \}. Exercise 6.1.1 reads as follows: ------------------------------------------------ "Show that the units of \mathbb{Z} are \ \pm 1, \ \pm i \ ."...

Solve the equation in the set of integers: $(a^2-b^2)^2=1+16a$

Find the number of integers $k$ with $1<k<2012$ for which there exist non-negative integers $a,\,b,\,c$ satisfying the equation $a^2(a^2+2c)-b^2(b^2+2c)=k$. ($a,\,b,\,c$ are not necessarily distinct.)

Total no. of positive integers ordered pairs of the equation 3^x+3^y+3^z = 7299

Given a three-digit integer $n$ written in its decimal form $\overline{abc}$. Define a function $d(n) := a + b + c + ab + ac + bc + abc$. Find, with proof, all the (three-digit) integers $n$ such that $d(n) = n$.

the problem: In how many ways can we write the number 4 as the sum of 5 non-negative integers?I realize this is a generalized combinations problem. I can plug it in using a formula, but I want to understand the logic behind why the generalizaed combination formula works. More specifically, my...