Integers Definition and 474 Threads

\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...

well this is the question... if a,m and n are positive integers with m<n, then (a^(2^m)+1) is a divisor of (a^(2^n)-1)... I started using induction and it works for the first step... but for the second one i do not know if i can make induction on m... any hint would help.. thanks :)

Homework Statement Prove that they are no integers a,b,n>1 such that (a^n - b^n) | (a^n + b^n). Homework Equations The Attempt at a Solution Do I solve this by contradiction? If so, how do I start it?

I saw this somewhere, it looks like fun but i can't seem to answer it integers: 1,2,3,4,5,6,7,8,9 \frac{a}{bc}+\frac{d}{ef}+\frac{g}{hi}=1 what is a,b,c,d,e,f,g,h,i ? pick from the above integers. (ONLY USE EACH OF THE ABOVE INTEGERS ONCE) :) bc means for example 35 (b=3 and c=5)...

Homework Statement if a is an algebraic number satisfying a^3+a+1 = 0 and b is an algebraic number satisfying b^2+b-3 = 0 prove that both a+b and ab are algebraic Homework Equations The Attempt at a Solution a is root of equation x^3+x+1 = 0 and similarly b, so there exists a x = ab,and...

Homework Statement Theorem Let \alpha\neq0 and \beta be Gaussian integers. Then there are Gaussian integers \tau and \rho such that \beta=\tau\alpha+\rho and N\left(\rho\right)<N\left(\alpha\right) Problem Show that the Guassian integers \tau and \rho in the Theorem are unique if and only...

Homework Statement find all orderde pairs of integers (x,y) such that x^2+y^2=4x+2y Homework Equations The Attempt at a Solution rearrange to--> x^2=4x+2y-y^2 because x and y can only be integers, y(2-y) must be divisible by x so y(2-y)>=x y(2-y)=x(x-4) x(x-4)>=x x-4>=1 x>=5...

Homework Statement If \omega is and nth root of unity, define Z[\omega], the set of generalized Gaussian integers to be the set of all complex numbers of the form m_{0}+m_{1}\omega+m_{2}\omega^{2}+...+m_{n-1}\omega^{n-1} where n and m_{i} are integers. Prove that the products of generalized...

Homework Statement Show that the infinite cyclic group Z is the unique group that is isomorphic to all its non-trivial proper subgroups Homework Equations The Attempt at a Solution Due to the fact that Z is cyclic and that every subgroup is a cyclic group, every subgroup of Z is a...

ok. some rant about definition and semantics. integers are isomorphic ordered pairs of natural numbers (a,b) w/ equivalence relation (a,b)=(c,d) iff a+d=b+c. reals are convergent sequences of rationals, etc. in mathematics, are integers simply isomorphic to the ordered pairs of...

Homework Statement Find a formula fo the sum of the fourth powers of the first n positive integers n ∑ i^4 (i=1) Justify your work using mathematical induction Homework Equations so i know the formula for the sum of the cubes of the first n positive integers k=n+1 ∑...

1. Find the sum of the odd integers greater than 15 but less than 241. a. 14,336 b. 28,672 c. 14,448 d. 28896 2. an = a1 + (n-1)*d 3. I know that n = 8 and a1= 17 and d = 2. But I don't know how to get one of these answers: 14,336 28,672 14,448 28896

How do we show the one point compactification of the positive integers is homeomorphic to the set K={0} U {1/n : n is a positive integer}? Say Y is the one point compactification of the positive integers. I know Y must contain Z+ and Y\Z+ is a single point. Also Y is a compact Hausdorff...

I was wondering what Q/Z, or the rational numbers modulo the integers. I am struggling to visualize what the cosets may be. Thank you for your time.

Homework Statement How many integers from 0-9,999,999 have exactly two 3's and two 5's as digits. Homework Equations I'm not really sure... The Attempt at a Solution The answer is 107520, if I'm not mistaken. I made a program to count it up for me, so I'm fairly sure that that is...

I was just wondering. the integers 1 2 3 are consecutive. are the integers 3 2 1 consecutive ?

Homework Statement Prove that for any two integers a and b, a+b=b+a. You may use the face that this holds for natural numbers. Homework Equations The Attempt at a Solution a=(x,y), b=(u,v) x,y,u,v are natural numbers a+b = (x,y)+(u,v) = (u,v)+ (x,y) = b+a I'm not sure if my...

Homework Statement How many 4-permutations of the positive integers not exceeding 100 contain three consecutive integers k, k+1, k+2, in the correct order: a) where these consecutive integers can perhaps be separated by other integers in the permutation? b) where they are in...

Homework Statement The sum of the reciprocals of two consecutive odd integers is 28/195. Find the two integers. Homework Equations ? The Attempt at a Solution 28/195= 1/x + 1/(x-2) Solved using quadratic formula and got 3.44, which does not seem to be right.. Any help?

Homework Statement If a is an algebraic integer with a^3 + a + 1 = 0 and b is an algebraic integer with b^2 + b - 3 = 0, prove that both a + b and ab are algebraic integers. Homework Equations An algebraic number is said to be an algebraic integer if it satisfies an equation of the form...

Hi, i have just registered to the forum, because this time i study number theory and in some problems i can't figure out how to solve them. This time i have to prove: If two integers x,y doesn't divided with 3 then the (x^2 - y^2) always is divided with 3. Does anyone has a clue how to...

Hi - I am trying to find the probability of hitting one of two boundaries in a simple random walk (I describe the problem precisely below). Actually, my main concern is to find the probability distribution over time to hit either one of two boundaries. I think that this is a very standard...

The rest of this sheet of problems was a piece of cake, and I think this is meant to be one of the easier problems on it, but I'm not quite sure how to do it. Homework Statement Find a fourth order polynomial with integer coefficients for which 1+\sqrt{5}-2\sqrt{3}The Attempt at a Solution I...

Given this permutations {1,2...,n}, prove that the directions of 1 and 2 never change. Proof: When generating permutations, one starts with everything having a left facing arrow. In order to determine what is mobile, the arrow must be pointing towards a smaller integer. 1 points to nothing...

Determine all possible positive decimal integer(s) P = X1X2X3….Xn, where P>=2 with none of the digits in P being zero, that satisfy this equation: P = X1^X1 + X2^X2 + ……+ Xn^Xn (For example, P = 234 cannot be a solution since 2^2 + 3^3 + 4^4 is equal to 287, not 234.) Notes: (i)...

Hello, Can an integer always be represented through the multiplication of two or more integers? (Are all integers divisible by some set of 2 or more integers (- or +)?) For example, 8 is can be represented by 1 x 8, 2 x 4 and 2 x 2 x 2. But what about 257 or even - integers? I'm trying...

How can i find all integers to this equation? 3(x+y)-xy=0 I allready found just by tring that if x=y then its 6 and x=12 ,y=4 (x=4 and y=12) I think that there are more, but how can i find all of them? PS. the first task was: 1/x + 1/y = 1/3 and then i had to find all integers...

Is the set of integers Z={0,+-1,+-2,...}... Hi, Can anybody help me. I know that integers under Addition is a group, but Is the set of Integers Z={0,+-1,+-2,...} together with the operation of subtraction a (noncommutative) group. Thanks a lot Flor

Homework Statement Hey, sorry about this. Its really obvious, i think there's just some really simple way to do it. Its annoying me, my teacher couldn't do it either! Evaluate 6667²-3333² (without a calculator) Homework Equations The Attempt at a Solution I know there is just...

i've come across a question thatr reads x^3 + 2y^3 + 4z^3 =0 show that x y z are all even part 2 requires to show that there are no such intergers i have no idea at all how to show something is even can anyone help please thanks

Homework Statement is the set of integers open or closed Homework Equations The Attempt at a Solution I thought not closed open because R/Z=Union of open intervals like ...U(-1,0)U(0,1)U...

Homework Statement Let N = AB, where A and B are positive integers that are relatively prime. Prove that ZN is isomorphic to ZA x ZB. The attempt at a solution I'm considering the map f(n) = (n mod A, n mod B). I've been able to prove that it is homomorphic and injective. Is it safe to...

Determine all pairs (x, y) of integers such that 1 + 2^x + 2^(2x+1) = y^2. ^=Exponent

Integers. I need help on this one please? Hi, i am trying to analyze the following problem, but i am new at abstract algebra so i am not sure whether i am reasoning properly. Problem: Why are there no integers x and y with x^2-y^2=34. Here is how i am reasoning: x^2-y^2=(x-y)(x+y)=34...

I am working on a problem and encountered the following problem: Given a,b element of {1,7,11,13,17,19,23,29} and also given that : x,y element of N+{0}. Now I want to *formlulate* the numbers that are _not_ reachable by the equation : z = ax + by + 30xy The formula(tion) should tell...

Let a be a positive integer. Find all positive integers n such that b = a^n satisfies the condition that a^2 + b^2 is divisible by ab + 1. Obviously if a=1 then all n work. Otherwise, we have a^2 + b^2 = a^2 (1+a^{2(n-1)}). Also, a^2 and a^{n+1} + 1 are relatively prime, so we need to find all...

I'm trying to search for duplicate values in a array of integers in Java. The array of intergers is sorted. Could anyone give me an idea on how to get started. :confused:

1.From the problem statement Let Z+ be the set of all positive integers; that is, Z+ = {1,2,3,...} Define Z+ x Z+ = {(a1, a2) : a1 is an element of Z+ and a2 is an element of Z+ } 2. If S is contained in Z+ and |S| >=3, prove that there exist distinct x,y that are elements of S such...

Homework Statement a. Prove that x^2+1 is irreducible over the field F of integers mod 11. b. Prove that x^2+x+4 is irreducible over the field F of integers mod 11. c. Prove that F[x]/(x^2+1) and F[x]/(x^2+x+4) are isomorphic. Homework Equations A polynomial p(x) in F[x] is said to...

Just have this question I am having trouble with The least common multiple of positive integers a, b, c and d is equal to a + b + c + d. Prove that abcd is divisible by at least one of 3 and 5. Thanks

Homework Statement I need to find some huge numbers and don't know how to do it using a computer. Everything that I have tried doesn't work because the numbers are too big. Homework Equations The equations are 1/((1-x)(1-x^2)(1-x^3)) and 1/((1-x)(1-x^3)(1-x^4)) where x=10^30. The...

Homework Statement prove:6 divides (n^3-n) for all integers n. Homework Equations n^3-n=(n)(n+1)(n-1) The Attempt at a Solution tried to use direct proof. Then used cases that involed n=2k for some integer k and n=2k+1 for some integer k. However, i could not get it so that 6 was...

Homework Statement Prove that for all x there exists and x if it is an element of the positive integers, then there is an integer y and an integer z. Homework Equations The Attempt at a Solution I know that the contrapositive would be "If there is not an element of the positive...

I have a solution to a problem which I am not certain that is complete. (It's a putnam problem so I can't believe I solved it) Would you mind to take a look at it? The problem stated: "Let A be any set of 20 distinct integers chosen from the arithmetic progression 1,4,7,...,100. Prove that...

[SOLVED] Set of n+1 integers from 2n [deleted by user]

Hi, I am just wondering if the zero product property (ab=0 implies a=0 or b=0) can be proven on the integers, or is it directly axiomatic to the defining of the integers? Also, where might I find a definition of the integer axioms? Thanks a lot,

Homework Statement I want to know what's the formula to calculate the sum of the square root of integers from 1 to n. I got an identity from wikipedia but its too complicated for me, it involves bernoulli's number, i don't know what is that. Homework Equations The Attempt at a...