Integer Definition and 620 Threads

Find the least positive integer $k$ such that $\displaystyle {2n\choose n}^{\small\dfrac{1}{n}}<k$ for all positive integers $n$.

Hey! :o I have to show that the number $a=201340168052123987111222893$ is not a square of an integer, without doing calculations.Could I solve this in $\mathbb{Z}_8$? I mean that the number $a$ can be written as followed: $$a=3+9 \cdot 10 +8 \cdot 10^2 + 2 \cdot 10^3+...$$ Since at...

Can some one explain to my why an integer equation that starts with 1 has a -1 at the end of the equation. example: 1 + 2 + 4 + 8 + 16 ... + 2 ^ N = 2 x ( 2 ^ N ) - 1 Conceptually where does the rule come from that there is a minus at the end of the equation. It starts with an odd number...

Show that $\displaystyle \sum_{k=0}^{2013} \dfrac{4026!}{(k!(2013-k)!)^2}$ is the square of an integer.

Find all integer solutions of the system $a^{a+b}=b^{12}$ $b^{b+a}=a^3$

So I think I've just proven a preposition, where ##0## is divisible by every integer. I prove it from the accepted result that ##a \cdot 0 = 0## for every ##a \in \mathbb{Z}##. From then, we can just multiply the result by the inverse of ##a##, to show that the statement holds for ##0##. That is...

FORTRAN error "array bound is not scalar integer" I'd like to know if a loop can be created, inside which I can call a subroutine in which there are arrays to be defined whose size varies as a function of loop variable. I tried as following, but got error "array bound is not scalar integer"...

In his book, Spivak did the proof by using the distributive property of integer. I am wondering if this, I think, simpler proof will also work. I want to show that ##a \cdot 0 = 0## for all ##a## using only the very basic property (no negative multiplication yet). For all ##a \in...

Your calculator tells you that the quantity $y=(2\cdot\sqrt[3]{2}+1-\sqrt{12\cdot\sqrt[3]{2}-15})^3$ is approximately an integer. Is $y$ exactly an integer? Justify your answer.

$$w = \frac{(ab - d) }{c - a - b}$$ I have to solve the above equation for variables `c` and `d` if `w` can be any number from $$w \in (-\infty, +\infty)$$ If we set `w = 0, then w = 1` we can solve for `c and d` $$0 = ab - d$$ $$d = ab$$ $$c = a + b$$ Now if I can substitute the values to...

Let $p, q, r, s \in \mathbb{N}$ such that $p \ge q \ge r \ge s$. Show that the function $f(x)=x^4-px^3-qx^2-rx-s$ has no integer root.

Hi, please could someone explain the notion that particles of 1/2 integer spin do not look the same when turned through 360 degrees. This notion seems to crop up when I read around QM but nobody seems to explain how this came about. So my question is this - what experiment shows/confirms that...

If $3^{n} +81$ is a perfect square, find a positive integer value of $n$. My Trail:: When $n\leq 4,$ then easy to know that $3^{n} +81$ is not a perfect square. Now let $\displaystyle n = k +4 (k\in \mathbb{Z^{+}}),$ then $3^{N} +81 = 81 (3^{k} +1).$ So $3^{N} +81$ is a perfect square...

Hi MHB, I don't know how to solve the problem below because I found what we are asked to prove is a bit confusing to me and hence I don't know how to formulate a credible way to solve the problem correctly. Problem: Let $s\ne t$ be real numbers. The equation $(x^2+20sx+10t)(x^2+20tx+10s)=0$...

Let $k$ be the largest root of $x^4+1-2009x=0$. Find the nearest integer to $\dfrac{1}{k^3-2009}$.

I have stumbled upon an approximation to the average of integer square roots. \sum^{n}_{k=1}{\sqrt{k}/n} \approx \sqrt{median(1,2,...,n)} Sorry I am not very good at LaTeX, but I hope this comes across okay. Could anyone explain why this might be happening? In fact, I just discovered that...

Let a sequence be defined as follows: $b_1=3$, $b_2=3$, and for $n \ge 2$, $b_{n+1}b_{n-1}=b_n^2+2007$. Find the largest integer less than or equal to $\dfrac{b_{2007}^2+b_{2006}^2}{b_{2007}b_{2006}}$.

Hello, I was wondering if someone knew the equation which describe the exact or statistical relation between the length(# of digits) of a power of 2 based on it's power. I plotted 200 of the powers in mathematica and I get a fairly straightforward staircase plot. I'm just wondering what's the...

Hi MHB, This problem drives me crazy because first, I have not encountered a problem like this before (but this is only an excuse and it must be my incompetence that holds me back from cracking it successfully) and I called off the attempt because I don't think I can solve it. Problem...

Given $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_0, a_a,\cdots,a_n$ are all smaller than 4 and are integer, $a_n \in (0, 1, 2,\cdots)$. Given that $f(4)=2009$, find $f(1)$.

no. of integer ordered pairs of $(x,y,z)$ in $ \sqrt{x^2-2x+6}\cdot\log_{3}(6-y) = x $ $ \sqrt{y^2-2y+6}\cdot\log_{3}(6-z) = y $ $ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x) = z $ My approach :: Here $6-x,6-y,6-z>0$. So $x,y,z<6$ Now $\displaystyle \log_{3}(6-y) =...

Find all solutions to $(m^2+k)(m+k^2)=(m-k)^3$, where $m$ and $k$ are non-zero integers.

1. Homework Statement Provide a simple formula or rule that generates the terms of an integer sequence that begins with : * 2,4,16,256,65536,... 3. The Attempt at a Solution I have tried a lot to solve it but i ended up with nothing,although i know that finding a term in the...

Find the smallest positive integer $n $ for which $n^{16}$ exceeds $16^{18}$.

For the integers $a, b, c$, we have $\dfrac{2a-b}{c}=\dfrac{2b+c}{a}=\dfrac{-2a-c}{b}$. If $a+b=2004$, find $a+b+c$.

I have a question about deriving the Bessel function of the second kind with integer order. I understand that the Bessel function and the second independent variable is defined as: L(y)=x^2y''+xy'+(x^{2}-n^{2})y=0 y_{2}(x)=aJ_m(x) ln(x)+\sum_{u=0}^{\infty} C_{u} x^{u+n} and Bessel first kind...

Calculation of Total no. of positive integer ordered pairs $(n,r)$ in $\displaystyle \binom{n}{r} = 120$ My Try:: Clearly $\displaystyle \binom{n}{r} = 120 \Rightarrow \binom{120}{1} = \binom{120}{119} = 120$ So $(n,r) = (120,1)\;\;,(120,119)$ are positive integer ordered pairs which satisfy...

Define $\{a_n\}$ is integer sequences (all term are integers) satisfy condition $a_n=a_{n-1}+\left\lfloor\dfrac{n^2-2n+2-a_{n-1}}{n}\right\rfloor $ for $n=1,2,...$ *note: $\left\lfloor x\right\rfloor$ is a greatest integer number less than or equal $x$ Find general term of sequences.

The no. of positive Integer ordered pair $(a,b)$ in $4^a+4a^2+4 = b^2$

I come across some odd stuff online... ... OK there's a typo for the 5 ... should be ##\small [\sqrt{9}]!-(9/9)## and the one for 7 looks a bit forced... What I'm wondering is if there are other sets that do something like this ... i.e. so for a given integer Z, we can find another integer...

$\text{given } :x \in\mathbb{Z}$ $f(x)= \begin{cases}x-3 & x \geq 1000 \\f\big [f(x+5)\big ]& x<1000 \end{cases} $ $\text{find } :\,\, f(84)$

C: "warning assignment makes integer from pointer without a cast" 1. I am trying to assign a color to the variable choice if it is equal to one of the 3 input numbers. if(pred==1) { choice = "RED"; } else if(pred==2) { choice = "GREEN"; } else if(pred==3) { choice =...

Determine the smallest integer that is square and starts with the first four figure 3005. Calculator may be used but solution by computers will not be accepted.(Tongueout)

I'm assuming this is a simple proof? I started thinking about it. It suffices to show that every even number can be written as positive powers of 2 (since every odd number is simply an even number plus 20). So let n = 2k for some non-negative integer k; we need to show that 2k = 2c1 + 4c2 +...

Is this line of thought correct? Please correct me where I´m wrong. Will this way of finding prime factors work when A is any integer? Is there a proof for this or a proof that is closely related? Is there a way to do it that requiers less iterations? It has to be a method that requires...

PS. I don't speak or write english very well. I´m doing my best. Is it still ok to post questions? a = any integer b = any prime number a * b = c Is there a proof that "c" isn't made up by any other prime factors than the prime factors that make up "a" (except for "b")?

I am curious, are there any calculators that calculate integer partitions with the stipulation that the calculator only calculate unique partitions. For example, if I want to calculate the number of ways to sum to 5 using 3 integers I have the following unique sums: 1 1 3 1 2 2 I would...

Find the smallest positive integer $n$ such that for every integer $m$ with $0<m<1993$, there exists an integer $k$ for which \frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}.

Find all pairs $(p, q)$ of integers such that $1+1996p+1998q=pq$.

Let $f(x)$ be a polynomial with integer coefficients. Show that $f(n)$ is composite for infinitely many integers $n$. EDIT: As Bacterius has pointed out we need to assume that $f(x)$ is a non-constant polynomial.

Homework Statement Prove that n! for n>1 cannot be a square or cube or any other power of an integer. Hint: There is always a prime between n/2 and n if n>3 The Attempt at a Solution Lets assume for contradiction that n!=x^r where x and r are natural numbers and n>3 , so there is some prime...

I'm trying to work with big integers but for some reason this program won't compile: program prob003 implicit none integer, parameter :: k32 = selected_int_kind(32) integer(kind=k32) :: num = 600851475143 end program prob003 The file name is prob003.f90 and I'm trying to...

For what integers p and q is where x=\sqrt {29}+\sqrt {89} is a root of the equation x^4+px^2+q=0

Question: Given a non-negative integer N, show many sets of non-negative integers (a,b,c,d) satisfy 2a+b+c+d=N Proposed (and roadblocked) solution: Case 1: 2a=0 Then there are \binom{N+2}{2} solutions (easy to prove). Case 2: 2a=2 Then there are \binom{N+2-2}{2} solutions. Case 3: 2a=4...

I can't seem to wrap my head around the types of sums used in probability theory, and here is a classic example. Section 6.1 of this article: http://en.wikipedia.org/wiki/Expected_value#Discrete_distribution_taking_only_non-negative_integer_values The first line of the proof, what is going...

I'm writing a MIPS assembly program and I'm trying to figure out a way to display a given integer in either decimal or binary using only one function with two parameters. The function takes in two parameters: an integer and a base (either '2' or '10'). I'd like to not edit anything other than...

$n\in N$ $100\leq n\leq 1997$ $\dfrac{2^n+2}{n} $ also is an integer please find n

Homework Statement Hey all. Not super familiar with using factorials, however, they do pop up occasionally. I understand that n! = 1*2*3*...*n. How do we treat factorial when we are multiplying n by an integer before taking the factorial? I know the answer for expanding (2n)!, however, I do...

Homework Statement We were asked to solve it using Augmented matrix. I just need one more equation though. A jar of coins contains only dimes, nickels, and pennies. There are 98 coins in the jar, and the total value of the coins is $6.49. Set up the system of equations representing this...

Find all positive integer solutions of the equation $4x^3-12x^2+5x-10y+36y^2-18y^3+4x^2y+6xy-15xy^2=0$.