Between which two consecutive integer numbers is $\sqrt{2}+\sqrt{3}$? My thoughts: $\sqrt{2}$ is $\approx1,4$ and $\sqrt{3}$ is $\approx{1,7}$ so their sum must be of the interval $(3;4)$. Any more strict approaches? Thank you in advance!
77804
How prove that $ \sqrt{2}+\sqrt{3}&gt;\pi$? Maybe some easy way?
910
I stumbled across this relationship while I was messing around. What's the proof, and how do I understand it intuitively? It doesn't really make sense to me that the sum of odd numbers up to $2x + 1$ should equal $x^2$.
110389
Prove that $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational.   I would assume that $\sqrt[31]{12} +\sqrt[12]{31}$ is rational and try to find a contradiction.  However, I don't know where to start. Can someone give me a tip on how to approach this problem?
73502
Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 &lt; i_2$.  Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$?  How would you construct such a rational number?  [I posted this only so that the useful answers at  could be merged here before that question was deleted.]
69195
Prove that for any set of $2n+3$ integers from the interval $[-2n-1,2n+1]$ there is a triple $(x,y,z)$ such that $x+y+z=0$.    Example  : Choose $5$ number from $\{-3,-2,-1,0,1,2,3\}$ there are $x,y,z$ with $x+y+z=0$.
31441
How can I find all integers $x≠3$ such that $x−3|x^3−3$?  I tried expand $x^3−3$ as a sum but I couldn't find a way after that.
87485
Is it true that all numbers of the form $\sqrt[b_1]{a_1}+\sqrt[b_2]{a_2}+\cdots+\sqrt[b_k]{a_k}$ are not integers, where $a_i,b_i$ are positive integers such that none of the $k$ terms is an integer? I think this must be a theorem somewhere, probably a fundamental theorem in number theory, but not sure what it would be called.
110294
Is it true that the product of $n&gt;1$ consecutive integers is never a $k$-th power of another integer for any $k \geq 2$?  I can see this is true in certain cases. For instance if the product ends on a prime, But how would one prove this in general?  Thanks for any help or suggestions.
77954
Let $a$ and $b$ be positive integers. Show that $\sqrt{2}$ always lies between $\dfrac{a}{b}$ and $\dfrac{a+2b}{a+b}$.  Please give the easy solution as possible.
49227
Prove that for each $n$, there exist $n$ consecutive integers, each of which is divisible by a perfect square larger than $1$. Can any one help with this? I was stuck because I know how to solve for a particular $n$ value, but I was helpless in the generalized form.
105141
How many subsets does the set $\{1, 2, \dots , n\}$ have that contain no two   consecutive integers if 1 and n also count as consecutive?   It looks that the number of such subsets obeys the (Fibonacci-style) formula $$ F(n) = F(n-1)+F(n-2) $$ with $F(0)=1$ and $F(1)=2$, but I wasn't able to prove this conjecture.
101083
Possible Duplicate:          I know how to prove $\sqrt 2$ is an irrational number. Who can tell me that why $\sqrt 3$ is a an irrational number?
54300
Yeah so I'm kind of stuck on this problem, and I have two questions.   Is there a way to define a number mathematically so that it cannot be a multiple of $7$? I know $7k+1,\ 7k+2,\ 7k+3,\ \cdots$, but that will take ages to prove for each case.  Is this a proof? $$(7k + a)^3 \equiv (0\cdot k + a)^3 \equiv a^3 \bmod 7$$    Thank you.
86086
I want to show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$.  I think it would be easier to prove it using the following: $\mathbb{Q}\subset\mathbb{Q}(\sqrt{3})\subset\mathbb{Q}(\sqrt[4]{3})$. Then $\sqrt{2}\notin\mathbb{Q}(\sqrt{3})$ so $\sqrt{2}\in \mathbb{Q}(\sqrt[4]{3})\Longleftrightarrow \sqrt{2}=a+b\sqrt[4]{3}$, with $a,b\in\mathbb{Q}(\sqrt{3})$.   I tried simplifying the above equation, but I did not get anything.
100958
Show that $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ is rational.   I've tried to attempt algebra on this problem. I noticed that there is some kind of nesting effect when trying to solve this. Please help me to understand how to attempt to denest this number.  Any help would be greatly appreciated.
109068
Find all positive integers $n$ such that $n$ is divisible by all the positive integers less than or equal to $\sqrt{n}$  My thought: If n is a positive integer, let d(n) denote the number of positive divisors of n and I will find n such that $d(n) \ge 2\sqrt{n} - 1$
18228
Let $X=\{1,2,3,\dots,n\}$ where $n\in N$. Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by   $\binom{n-r+1}{r}$ where $0\le r\le n-r+1$.  I am unable to think of any possible way to prove it.  Please explain the proof and comment on the technique used to help me with problems in the future.   Thanks in Advance!  Thanks for pointing the typing faults!