Integer Definition and 620 Threads

Hello I am trying to simplify a set of equations to something easier to work with. For example 11x+4 13x+11 31x+17 19x+2 14x+9 Over integer values for x, I need to find numbers that won't be hit by these equations. The first one hits 4,15,26,37,48,59 for x=0,1,2,3,4,5. Is there a way to...

Hi, does anybody know of any good sources to learn about the Integer Quantum Hall effect from the perspective of theoretical physics? Any suggestion will be appreciated, thanks.

Given is the function f: R2 -> R, with f(x,y)=x2+y2-6xy+8y. The level surface f(x,y)=1 contains infinitely much points (x,y) where x and y are integer. How can I prove this? I see that it is true with some examples, but how can I prove this. Do I need to use the gradient? Or tangent planes? Or...

I'm not sure how to do these problems here. Help me learn them please!

Determine the positive numbers, $a$, such that the sum: $$\sqrt[3]{3+\sqrt{a}}+\sqrt[3]{3-\sqrt{a}}$$ is an integer.

Given a rational number, $\frac{p}{q}$, show that there are only a finite number of positive integer solutions to the equation: $$\frac{1}{x}+\frac{1}{y}=\frac{p}{q}$$

What an innocently looking equation. If we allow negative integers, a=4, b=-1, c=11 is a solution. Do some tricks with divisibility? Solve for a? Brute force with the computer?It won't help. There are solutions, but the smallest solution has 80 digits. What happens if we replace 4 by other...

Prove, that the equation \[x_1^3+x_2^3+x_3^3+x_4^3+x_5^3 = n\] has an integer solution for any integer $n$.

I am trying to understand how to find solutions for a problem when parameters are limited to positive integers. Example: 30x+19= 7y+1 =a ; where x,y,a are positive integers Wolframalpha outputs: a = 210 n + 169, x = 7 n + 5, y = 30 n + 24, n element Z(integers) 30*7= 210 (obviously) How do I...

I recently came across a problem in Irodov which dealt with the gravitational field strength of a sphere. Took some time to get my head around it and figure how to frame a triple integral, but it felt good at the end. Am I going to start seeing triple integrals in the freshman year tho? If so...

Homework Statement Homework Equations none The Attempt at a Solution i assumed it can be factored into the form ## (x^2 + m_1 x + m_0)(x + n_0) ## by comparison of coefficients ## m_0 n_0 = -abc -1\\ m_1 + n_0 = -a -b -c\\ m_0 + m_1 n_0 = ab +ac + bc\\ ## the only other information i have is...

Homework Statement [/B]Homework EquationsThe Attempt at a Solution i tried to do it by writing it as ## a_{1999} x^{1999} + a_{1998} x^{1998} ... a_0 \pm1 = 0 ## for 1999 different integer values of x i am thinking of writing it as ## a_{1999} x^{1999} = -a_{1998} x^{1998} - a_{1997} x ^...

Hi, to understand finally the Laue equation for diffraction I am missing something : h*p+k*q+l*r = integer. Given that p,q,r are integers how come h,k,l MUST BE INTEGERS as well? Say p=q=r=2, than h=k=l=1/2 works just fine. I understand that there is something about a common...

I have studied the integer quantum hall effect mainly from David Tong's notes and i understand how the ## \rho_{xy}## is quantized in terms of the chern number. What I don't understand is - how the chern numbers relate to the number of filled Landau levels though. - I also don't understand the...

Homework Statement let p(x) be a polynomial with integer coefficients satisfying p(0) = p(1) = 1999 show that p has no integer zeros Homework EquationsThe Attempt at a Solution ## p(x) = \sum_{i= 0}^{n}{a_i x^i} ##[/B] using the given information a0 = 1999( a prime number) and ## a_n +...

Is anybody familiar with any theory of integer cevians on equilateral triangles? More specificaly, I was trying to find something about the number of integer cevians that divide the side in integer parts. Like, the eq triangle of side 8 have cevian 7 dividing one side into 3+5. Only reference...

I have done it by the parametric form of σ, but if I change σ to implicit form that is G(x,y,z)=x^2+y*2+z^2-R^2=0 I don't know how continue. The theory is: where Rxy is the projection of σ in plane xy so it's the circumference x^2+y^2=R^2

Show that there are infiinite isosceles triangles which have integer sides and integer areas

Homework Statement Let ##S = \{\frac{1}{n} + \mathbb{Z} ~|~ n \in \mathbb{N} \}##. I am trying to show that ##f : \mathbb{N} \rightarrow S## defined by ##f(n) = \frac{1}{n} + \mathbb{Z}## is a bijection. Surjectivity is trivial, but injectivity is a little more involved. Homework EquationsThe...

$x^2+y^2=1992---(A)$ pove $(A)$ has no positive integer solution

Prove, that the equation:$x^3+y^3+z^3 = 2$- has infinitely many integer solutions.

$$a\,\, sequence:a_1=\dfrac {1}{3}\,\, and \,\, \,\, a_{n+1}=a_n^2+a_n,\,\, n=1,2,3,----2000\\ find \,\, the \,\, integer \,\, part\,\, of :\sum_{n=1}^{2001}\dfrac {1}{a_{n}+1}$$

This might seem like a very simple problem, because we could just say that the only possible solution is n = 1/2, which is not an integer. But I am curious as to how to prove that there is no solution, with no knowledge of rational numbers, just as we can prove that x^2 = 2 as no rational...

$a,b\in N$ $k=\dfrac {ab^2-1}{a^2b+1}\,\, \,also \,\,\in N$ find pair(s) of $(a,b)$

I have authored documents of 40 years of computer software development with a mind to collect them into a publication at some point. They have been built around several software topics but mathemetics is a favorite of mine. I find a point of inspiration and write a piece of software around it...

Is there a function in Matlab that presets the value of a variable as an integer only? For example I will set the variable 'y' as an integer at the very beginning of the code, and whenever the variable gains a new value it automatically returns an integer value. Thank you in advance.

Hey guys, I have an assignment on fortran, which basically is supposed to read grades of a class and print them in alphabetical order and wether if the student failed or passed the class. We're supposed to do everything using our basic knowledge of fortran. When I run it it says 'integer...

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...

I have one pretty basic question... What are the types short, long etc used when creating for example integers used for? Whenever I read the description, it says "it's able to store bigger numbers", but to be honest I don't see any practical use of/visualize such a description... Why would I...

Hey! :o I want to calculate the limit $$\lim_{x\rightarrow \infty}x^{100}\left [\frac{1}{x}\right ]$$ When $x\rightarrow +\infty$ it holds that $0<\frac{1}{x}<1$, or not? (Wondering) If yes, it holds that $\left [\frac{1}{x}\right ]=0$ or not? Then $x^{100}\left [\frac{1}{x}\right ]=0$, and...

Homework Statement Suppose that a and b are odd integers with a ≠ b. Show that there is a unique integer c such that |a - c| = |b - c| Homework EquationsThe Attempt at a Solution What I did was this: Using the definition of absolute value, we have that ##(a - c) = \pm (b - c)##. If we choose...

Homework Statement Suppose that ##x \in \mathbb{N}## is such that ##n < x < 2n##, where ##n## is another natural number. I want to conclude that it is impossible for ##x## to divide ##2n##. Also, is there a way to naturally generalize this to all integers? Homework EquationsThe Attempt at a...

Homework Statement (a) Cobalt has only one stable isotope, 59Co. What form of radioactive decay would you expect the isotope 60Co to undergo? Give a reason for your answer. (b) The radioactive nuclei 21084Po emit alpha particles of a single energy, the product nuclei being 20682Pb. (b) (i)...

Let $a = 5^{1000}\cdot sin(1000\alpha)$ , where $sin(\alpha)=\frac{3}{5}$. Prove that $a\in \mathbb{Z},$ and find $a\:\: (mod \:\:5)$.

prove by induction for all positive integers n: 1+5+9+13+...+(4n-3)= n/2(4n-2) i tried this by trying to prove n/2(4n-2)+ (4(k+1)-3) = k+1/2(4(k+1)-2) but it did not work out for me.

I am stuck with one proof and I need some help because I don't have any idea how to proceed at this moment. The task says: If f(x) is a polynomial with integer coefficients, and if f(a)=f(b)=f(c)=-1, where a,b,c are three unequal integers, the equation f(x)=0 does not have integer solutions...

Consider a quantum scalar field theory with interaction terms of the form ##\phi^{n}##, where ##n## is not an integer. Where are some examples of physical theories which involve such interaction terms?

I have $$\sum_{n = 1}^{\infty} \frac{2^n}{n^{100}}$$ and I need to find whether it converges or diverges. I can use the ratio test to get: $$\lim_{{n}\to{\infty}} \frac{2^{n + 1}\cdot n^{100}}{2^n \cdot (n + 1)^{100}}$$ But I'm not sure how to get the limit from this. I know the limit of...

I couldn't find this problem anywhere else on the forum so I thought I'd post it. If however, I am duplicating, mods feel free to remove the post :p. No doubt many of you know it already, but I found it quite interesting. Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 +...

Homework Statement Prove by induction, that when r(r-1)(r+1) is an even integer when r=2,3,4... Homework Equations Prove by induction The Attempt at a Solution I began with the base case r=2, leading 6. Then I proceed with r=3, leading 24. Now if r=k is true, then k(k-1)(k+1) is also true...

Find all integers x such that $x(x+1)(x+7)(x+8)$ is square of an integer

I have a user input of 2 integers (m,n) Then my system will generate 1 list of M (m,n < M) integers that start at m and end at Mth integer of value xM. The formula to calculate xM is followed by x_0=m x_M=x_{M-1}+n After the list is generated I randomly delete N (N << M) rows from it and given...

Hi, Is there a way to formulate the solution of minimization of: abs(n-2^x*3^y) Over integers x and y for any given integer n? A numeric example that I found by trial and error is: |6859-2^8*3^3|=53 Thanks in advance.

Find all solutions in integers $x,y$ of the equation $y^2+2y= x^4+20x^3+104x^2 + 40x + 2003$

Hello, I have a question regarding "polynomials" that have terms with interger and fractional powers. Homework Statement I want to solve: $$ x+a(x^2-b)^{1/2}+c=0$$ Homework Equations The Attempt at a Solution My approach is to make a change of variable x=f(y) to get a true polynomial (integer...

I have been studying discrete mathematics for fun and I am kind of stuck on this bijection problem. 1. Homework Statement I wanted to apologize in advance if i put this homework question in the wrong part of the forums. Discrete Math and much logic math is a computer science type math of...

Homework Statement Let n be an odd integer ≥ 5. Find the number of triplets (x, y, z) of positive integers which satisfy the equation x + y + 2z = n Homework Equations Do not know The Attempt at a Solution Let n = 2k + 1, k ≥ 2 x + y + 2z = n 2z = 2k + 1 - (x + y) ≤ 2k + 1 - 2 (because x +...

Let $f(x) = Ax^2 + Bx +C$ where A,B,C are real numbers. prove that if $f(x)$ is integer for all integers x then $2A, A + B, C$ are integers. prove the converse as well.

I have been looking at various proofs of this statement, for example Proof 1 on this page : http://www.cut-the-knot.org/proofs/sq_root.shtml I'd like to know if the following can be considered as a valid and rigorous proof: Given ##y \in \mathbb{Z}##, we are looking for integers m and n ##\in...

A few months ago I posted a simple equation that shows an interesting nexus between the difference between the squares of successive integers and the sums of their roots, viz: Where y = x+1 then (x + y) = (y2 - x2) Recently I expanded this relationship as follows: Where n is any integer and y...