Integer Definition and 620 Threads

Let $P(x)$ be a polynomial with integral coefficients. Suppose that there exist four distinct integers $x_1,\,x_2,\,x_3,\,x_4$ with $P(x_1)=P(x_2)=P(x_3)=P(x_4)=5$. Prove that there is no integer $a$ with $P(a)=8$.

Hello! (Wave) Proposition: "$\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds $\bigcap_{n \in \mathbb{N}_0} p^n \mathbb{Z}_p=0$ and $\mathbb{Z}_p \ p^n\mathbb{Z}_p \cong \mathbb{Z} \ p^n \mathbb{Z}$. Especially $p\mathbb{Z}_p$ is the only...

f(n) is defined as 11x22x33...xnn Then it seems as if f(n)/(f(r).f(n-r)) is always an integer for 0 < r < n. I tried a few cases. Its true for them. Is it always true? I cannot seem to figure out any ways to prove it.

Prove that the equation $x_1^3+x_2^3+x_3^3+x_4^3+x_5^3=k$ has an integer solution for any integer $k$.

Greg Egan just proved something nice: the E10 lattice, famous in string theory and supergravity, can be described as the lattice of self-adjoint matrices with integral octonions as entries! I'm not sure this result is new, but I've been wanting it for quite a while and haven't seen it...

I'm unsure if this is a calculus or precalculus topic, but it's from a calculus book, so I'm putting it here. (Note \lfloor x \rfloor means the floor of x or the greatest integer less than or equal to x.) Prove that \lfloor x \rfloor +\lfloor y \rfloor \leq \lfloor x+y \rfloor \leq \lfloor x...

Hi everyone, so I was relooking into the external EEPROM problem I had earlier for the Arduino, and am thinking that I am missing even just a basic understanding of it. So I'm using this https://www.sparkfun.com/products/525 EEPROM which should have 256kbits of space. I am hoping to save 4...

Homework Statement A number divisible by both 16 and 15 is divisible by at least how many unique integers? Homework Equations Combination formula, nCk = n!/(k!(n-k)!) The Attempt at a Solution I found the prime factorization for 16 and 15, 2x2x2x2x3x5. A number divisible by 16 and 15 must be...

Hello! (Wave) I want to conclude that the number $\frac{1}{2}$ is an integer $5-$ adic and to calculate the first five positions of its powerseries. In order to conclude that $\frac{1}{2}$ is an integer $5-$ adic, do I have to use this definition? Let $p \in \mathbb{P}$. The set of the...

Hey! (Wave) Let the ring of the integer $p$-adic numbers $\mathbb{Z}_p$. Could you explain me the following sentences? (Worried) It is a principal ideal domain. $$$$ The function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p$ is an embedding. (So, $\mathbb{Z}$ is considered $\subseteq...

Prove by permutations or otherwise $\displaystyle \frac{\left(n!\right)!}{\left(n!\right)^{(n-1)!}}$, where $n\in \mathbb{N}$

Prove by permutations or otherwise $\displaystyle \frac{\left(n^2\right)!}{\left(n!\right)^n}$, where $n\in \mathbb{N}$

$\sqrt{\text{mbh}_{29}}$ Challenge: Sn = 3, 293, 7862, 32251, 7105061, 335283445, 12826573186, ?, ?, 44164106654163 S1 through S7 begin an infinite integer sequence, not found in OEIS. 1) Find S8 and S9. 2) Does S10 belong to Sn? 3) If S10 is incorrect, what is the correct value of S10...

Prove that if the sides of a triangle are prime numbers its area cannot be whole number.

Let $p,\,q,\,r,\,s$ be positive integers such that $p\ge q\ge r \ge s$. Prove that the equation $x^4-px^3-qx^2-rx-s=0$ has no integer solution.

Prove that $\dfrac{m^2!}{(m!)^2}$ is an integer, where $m$ is a positive integer.

The sum of three integers $x,\,y,\,z$ is zero. Show that $2x^4+2y^4+2z^4$ is the square of an integer.

Determine all positive integers $k$ for which $f(k)>f(k+1)$ where $f(k)=\left\lfloor{\dfrac{k}{\left\lfloor{\sqrt{k}}\right\rfloor}}\right\rfloor$ for $k\in \Bbb{Z}^*$.

Find an integer $k$ for which $\dfrac{1}{k}+\dfrac{1}{k+1}+\dfrac{1}{k+2}+\cdots+\dfrac{1}{k^2}>2000$.

Suppose $X$ is a number of the form $\displaystyle X=\sum_{k=1}^{60} \epsilon_k \cdot k^{k^k}$, where each $\epsilon_k$ is either 1 or -1. Prove that $X$ is not the fifth power of any integer.

Mathematicians have long held that infinite integers do NOT exist, but here is a very simple argument that shows that they do exist. A list of positive integers Z+ can be formed in a base-1 numeral system as... 1 11 111 1111 . . . 1111111... Since the set of integers is infinite...

Prove that $(-1,\,\pm 1)$, $(0,\,\pm 1)$, $(3,\,\pm 11)$ are the only integers solution for the equation $y^2=x^4+x^3+x^2+x+1$.

I tried to prove this claim, as I require it to finish one of my proofs. By definition, if a,b are integers, with a \ne 0, we say that a divides b if there exists an integer c such that b = ca. So, n divides 0 means that there exists some integer c such that n = c*0 = 0. But this would...

Obviously it will take some brute-force. But how do I minimize the brute-force needed (optimize)? I know one can solve Diophantine equations and quadratic Diophantine equations. But what if I have something like 10 (any number) of variables? (what if there are no squares, what if there are...

Can anyone help me prove the greatest integer function inequality- n≤ x <n+1 for some x belongs to R and n is a unique integer this is how I tried to prove it- consider a set S of Real numbers which is bounded below say min(S)=inf(S)=n so n≤x by the property x<inf(S) + h we have...

Ok guys, I know this must be pretty basic for but I am new to this section of physics. Anyway, my question is a two-part one, I guess: 1) Why does the spin number get only half integer values in fermions and integer values in bosons, mesons, etc.? 2) How do we conclude that the spin number is...

Show that if a polinomial $P(x)$ with integer coefficients takes the value 7 for four different integer values of x then there is no integer x for which $P(x) = 14$

Homework Statement Problem: Prove that every positive integer P can be expressed uniquely in the form P = a_0 * 2^n + a_1 * 2^(n – 1) + a_2 * 2^(n – 2) + . . . + a_n where the coefficients a_i are either 0 or 1. Solution: Dividing P by 2, we have P/2 = a_0 * 2^(n – 1) + a_1 * 2^(n – 2) + . ...

Find the greatest positive integer $x$ such that $x^3+4x^2-15x-18$ is the cube of an integer.

Find all integer solutions $(a,\,b)$ satisfying $a^4+79+b^4=48ab$.

Homework Statement in mod 35, find the inverse of 13 and use it to solve 13x = 9 gcd(35,13) =1 so the inverse exsists: 35 = 2*13 + 9 13 = 1*9 + 4 9 = 2*4 + 1 4 = 4*1 and then to find the linear combination 1 = 9 - (2*4) = 9 - 2(13-9) = 3*9 - 2*13 = 3* (35 - 2*13) - 2*13 = 3*35 - 8*13 =...

There is a vector ##v_i(t)## (i=1,2,3). How to define the three functions in Mathematica? What about ##t_{ij}(t,\vec{x})##? I am trying to solve my vector and tensor equtions with Mathematica. Analytical solution would be perfect but numerical solution would also be fine. Actually I am not...

Homework Statement The expression sin(2°) sin(4°) sin(6°)... sin(90°) is equal to a number of the form (n√5)/2^50 where "n" is an integer. Find n Homework Equations geometric sum: a/ 1-r The Attempt at a Solution I found the solution online but have no idea how they got it...

$a,b\in N ,\, and \,\, a>b,\,\, sin \,\theta=\dfrac {2ab}{a^2+b^2}$ (where $0<\theta <\dfrac {\pi}{2}$) $A_n=(a^2+b^2)^nsin \,n\theta$ prove :$A_n$ is an integer for all n $\in N$

show that the equation $4xy - x - y = z^2$ has no positive integer solution

Hi everyone, :) Here's a question I got from a mailing list in our university. Interesting problem but I sill have no clue on how to solve it. Note that this is a research level question and a solution may or may not exist.

Homework Statement For a set of vectors in R3, is the set of vectors all of whose coordinates are integers a subspace?The Attempt at a Solution I do not exactly understand if I should be looking for a violation or a universal proof. If x,y, z \in Z then x,y,z can be writted as...

solve in integers x, y, z(parametric form) $4x^2 + 9 y^2 = 72z^2$

Homework Statement The problem (and its solution) are attached in TheProblemAndSolution.jpg. Specifically, I am referring to problem (c). Homework Equations Set theory. Union. Integers. Prime numbers. The Attempt at a Solution I see how we have all multiples of all prime numbers in...

This question specifically relates to a numerator of '1'. So if I had the irrational number √75: 1/(x*√75) Could I have some irrational non-transindental value x that would yield a non '1', positive integer while the x value is also less than 1/√75? Caviat being x also can't just be a division...

If $x,y$ are integer ordered pair of $2x^2-3xy-2y^2 = 7,$ Then $\left|x+y \right| = $ My Try:: Given $2x^2-3xy-2y^2 = 7\Rightarrow 2x^2-4xy+xy-2y^2 = 7$ So $(2x-y)\cdot (x-2y) = 7.$ Now How can I solve after that Help me Thanks

Hi, I can't understand why the statement in the title is true. This is what I know so far that is relevant: - A subgroup of a cyclic group G = <g> is cyclic and is <g^k> for some nonnegative integer k. If G is finite (say |G|=n) then k can be chosen so that k divides n, and so order of g^k...

$A=(\dfrac{16\times72+17\times73+18\times74+19\times75}{16\times71+17\times72+18\times73+19\times74})\times 150$ find the integer part of A

If $P(x)=x^4+mx^3+nx^2+px+q$ is a polynomial whose roots are all negative integers, and given that $m+n+p+q=2009$, find $q$.

Show that the equation $a^2=b^4+b^2+1$ does not have integer solutions.

Find all pairs $(a,\,b)$ of integers such that $1998a+1996b+1=ab$.

$x,y,z,w $ are all integers if (1):$ w>x>y>z$ and(2) :$2^w+2^x+2^y+2^z=1288\dfrac {1}{4} $ find $x,y,z,w$

Find all values of $(a,\,b)$ where they are positive integers for which $\dfrac{a^2+b^2}{a-b}$ is an integer and divides 1995.

Find all integer solutions of the equation $(x^2-y^2)^2=1+16y$

"Prove ##5|(3^{3n+1}+2^{n+1})## for every positive integer ##n##." (Exercise 11.8 from Mathematical Proofs: A Transition to Advanced Mathematics 3rd edition by Chartrand, Polimeni & Zhang; pg. 282) I'm having difficulty solving this exercise. My first thought was to use induction but I get...