6
votes
3answers
106 views

Using decimals of $\pi$ to store data

I read recently about an idea to, instead of storing actual data, converting the data to a string of digits and then store the index of where this pattern occurs in some number, for example $\pi$. The ...
0
votes
2answers
49 views

How to find irrational approximates

Say I have a rational number, $n$, that approximates an irrational number of the form: $$n \approx {a+\sqrt b \over c}$$ in terms of being irrational. What is a good way of finding the unknown ...
1
vote
2answers
45 views

Does the absence of horizontal lines shows that there are no $n,m\in \mathbb{N}$ such that $n^2=2m^2$?

When I was learning about the proof of the irracionality of $\sqrt{2}$, I remember of trying to visualize it by ploting the graphs of $f(n)=n^2$ and $g(m)=2m^2$, but at the time I got confused and ...
2
votes
1answer
77 views

Does the limit of a sequence with floor function exist?

Question : Let $a_n=n\alpha-\lfloor n\alpha\rfloor\ (n=1,2,\cdots)$ where $\alpha$ is an irrational number. Then, does the limit $n\to\infty$ of $(a_n)^n$ exist? I know that ...
0
votes
3answers
45 views

A pretty much simple number theory problem

Let $x$ be an irrational number, and $n$ be a positive integer. Will there ever be a set of $(n,x)$ which satisfies $x(n-x) \in \mathbb{Z}$ ? If so, could you suggest those numbers? And, if not, ...
5
votes
2answers
172 views

A question about decimal representation of irrational numbers.

Is this true that any finite word of the alphabet $\mathcal{A_9}=\{0,1,2, \ldots,8,9\}$ appears somewhere in the decimal representation of $\sqrt{2}$ ? Thanks !
5
votes
1answer
54 views

$\lim \{r^n\}$ exists, Is $r$ an integer?

$r\in\Bbb R$, $|r|\gt1$ and $\lim\limits_{n\to\infty}\{r^n\}$ exists. Can one conclude that $r$ is an integer? Here, $\{x\}=x-[x] $ is the fractional part of $x\in\Bbb R$ If $r\in\Bbb Q$, the ...
6
votes
2answers
123 views

If $\sum\frac1{a_n}$ is convergent, then irrational?

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=1$$ If $\sum\limits_{n=1}^\infty\frac1{a_n}$ is convergent, can one conclude ...
17
votes
3answers
214 views

Is $\sum\limits_{n=1}^\infty\frac1{a_n}$ irrational?

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=+\infty$$ Can one conclude that $\sum\limits_{n=1}^\infty\frac1{a_n}$ is an ...
8
votes
1answer
122 views

$\sum\limits_{n=1}^\infty\frac1{a_n}$ is irrational

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{a_1a_2\dotsb a_n}=+\infty$$ then $\sum\limits_{n=1}^\infty\frac1{a_n}$ is an irrational ...
8
votes
2answers
226 views

Is $\ \Large\pi^e$ rational?

Is the number $\ \Large\pi^e$ rational?
0
votes
1answer
31 views

If $\frac1\alpha+\frac1\beta=1$, irrational, then $\{\lfloor n\alpha\rfloor:n\in\Bbb N\}\uplus\{\lfloor n\beta\rfloor:n\in\Bbb N\}=\Bbb N$

Let $\alpha,\beta\in\Bbb R\setminus\Bbb Q$ such that $\frac1\alpha+\frac1\beta=1$, and define $S(x)=\{\lfloor nx\rfloor:n\in\Bbb N\}$. (Note that my convention takes $0\notin\Bbb N$.) The claim is ...
1
vote
1answer
47 views

Rational vs irrational

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
1
vote
1answer
77 views

Dense sequence in $[0,1]$

There is the theorem proved by Liouville which states that if $x$ is irrational then there are infinitely many fractions $\frac{p}{q}$ such that $|x-\frac{p}{q}|<\frac{1}{q^2}$, i.e. ...
1
vote
0answers
60 views

Irrational numbers and series

Let $$f(x) = \prod_{n = 0}^\infty \left(1 + \frac{x}{2^n}\right)$$ According to an exercise in a packet of problems in elementary number theory, this function and all its derivatives are irrational ...
1
vote
3answers
333 views

Finding set of non recurring non terminating decimals

I need to find a set of two Integers P and Q such that ...
10
votes
1answer
167 views

Does $\lfloor(4+\sqrt{11})^{n}\rfloor \pmod {100}$ repeat every $20$ cycles of $n$?

I recently came across a post on SO, asking to calculate the least two decimal digits of the integer part of $(4+\sqrt{11})^{n}$, for any integer $n \geq 2$ (see here). The author presented a Java ...
8
votes
0answers
184 views

Irrationality of $\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper, the author recalls several sufficient ...
2
votes
2answers
144 views

Proof of irrationality of $\dfrac{\sqrt{8}}{\sqrt{7}}$

We have to prove that $\dfrac{\sqrt{8}}{\sqrt{7}}$ is irrational(try not to use the Rational Root Theorem) At first,we prove that the expression is not an integer. ...
6
votes
0answers
127 views

Does the number $2.3\,5\,7\,11\,13\ldots$ exist and, if so, is it rational or irrational &/or transcendental? [duplicate]

Does there exist a number which contains in its digits all of the prime numbers listed in order: $$2.3\,5\,7\,11\,13\ldots\ldots$$ if so, will it be rational or irrational &/or transcendental?
19
votes
2answers
268 views

Is there an elementary proof that $\sum_{n=1}^\infty {1\over n^s\{n\pi\}}<\infty$ for some $s>0$?

Edit: David Speyer's answer made me realize a couple of things and I would like to clarify. Sorry if the length of this is getting out of hand. First, it is now clear that no estimate can be obtained ...
0
votes
0answers
52 views

Solutions of $3a^2-2b^2=1$ [duplicate]

The question is to find the integral solutions of $$3a^2-2b^2=1$$ In the solution it is given that: The solutions of $3a^2-2b^2=1$ can be obtained from ...
3
votes
1answer
79 views

Irrational number?

Is the solution of the equation $$x + \arctan(x) = \pi$$ irrational ? The equation of $x + \arctan(x) = 1$ must be transcendental because for any nonzero algebraic $x$, $arctan(x)$ is ...
12
votes
1answer
351 views

Prove that this number is irrational

The number $a=0.12457...$ is defined as follows: The digit on the $n$-th place after the dot is the first digit left to the dot of the number $n\sqrt2$. For example, for $n=1$ we have ...
8
votes
1answer
75 views

Number made from ending digits of primes

Consider the number 0.23571379391713739171393971379371799173739113791379391173917133713717793 ... The number is formed by the ending digits of the prime numbers. Is it known whether this number is ...
9
votes
1answer
188 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $S(\alpha)=\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \}$. (Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers.) Question. Is ...
3
votes
3answers
445 views

Sum of all real number for any interval.

We know that sum of natural numbers over any interval always exists. For example sum from 0 to 10 of all natural numbers is $$S=\sum_{n=0}^{10}{n}=\frac{0+10}{2}\times{10}=55$$ But what about real ...
3
votes
2answers
60 views

Does $E^2 \; ( E \approx 1.2640847\ldots)$ equal $D \approx 1.5979102\ldots$?

Does $E^2=D$? Where $E$ is a constant used in the closed form of the Sylvester Sequence (see: Closed form formula and asymptotics) and $D$ is a constant for the closed formula of the sequence A007018 ...
2
votes
0answers
41 views

Experimental calculation and $\mathbb{Q}$

I have been reading this article and have a question about the first line of the second paragraph on the first page. It says: The basis for this suggestion is the simple fact that all experimental ...
-2
votes
1answer
238 views

Proof of $\pi$+$e$ irrational

The wikipedia tells that it is not known that $\pi+e$ is irrational? Immediately after reading this my mind came with this proof- Let $x =\sqrt{\pi^2}+\sqrt{e^2}$ be rational, then $ \quad ...
0
votes
1answer
199 views

Proving that any rational number can be represented as the sum of the each cube of three rational numbers

I found the following question in a book: Prove that any integer can be represented as the sum of the each cube of five integers. The answer : ...
11
votes
1answer
283 views

Linear independence of the numbers $\{1,\pi,{\pi}^2\}$

Does someone know a proof that $\{1,\pi,{\pi}^2\}$ is linearly independent over $\mathbb{Q}$ ? The proof should not use that $\pi$ is transcendental. $\{1,e,e^2,e^3\}$ is linearly independent over ...
26
votes
2answers
474 views

Linear independence of the numbers $\{1,e,e^2,e^3\}$

Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$? The proof should not use that $e$ is transcendental. $e:$ Euler's number. $\{1,e,e^2\}$ is linearly ...
1
vote
2answers
50 views

Integer outputs of $y=x^2$ , do their last digits form an irrational?

Let the domain of $y=x^2$ be the positive integers. I input consecutive positive integers from $[1, \infty)$ their last digits are $a, b, c, ...$ respectively. If I then make the number $z=\frac ...
3
votes
1answer
87 views

Is there any kind of irrational number wich does not contain digit 9?

At first we must prove that there is or is`t irrational numbers which does not contain digit 9! if there are many kind of such numbers, then there is another question: how to write down algebraic ...
0
votes
3answers
177 views

how to find out any digit of any irrational number?

We know that irrational number has not periodic digits of finite number as rational number. All this means that we can find out which digit exist in any position of rational number. But what about ...
1
vote
1answer
161 views

how do we know the BBP formula for $\pi$ is valid?

I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we ...
4
votes
0answers
141 views

Proof of $\pi$ not being a quadratic irrational number.

Does someone know a proof (books , articles) that $\pi$ is not a quadratic irrational? The proof should not use that $\pi$ is transcendental. Any hints would be appreciated.
1
vote
0answers
80 views

Gelfond-Schneider Constant $2^{\sqrt{2}}$

Someone knows a proof (books , articles) that $2^{\sqrt{2}}$ is irrational ? Without using that $2^{\sqrt{2}}$ is transcendent. Any hints would be appreciated.
0
votes
2answers
82 views

$\pi$ does not lie in any quadratic extension of $\mathbb{Q}$

Knowing that $\pi^2$ is irrational: How can we prove that $\pi$ does not lie in any quadratic extension of $\mathbb{Q}$ ? Without using that $\pi$ is transcendent. Any hints would be appreciated.
11
votes
1answer
375 views

Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?

Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? I doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = ...
87
votes
1answer
20k views

Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can ...
9
votes
3answers
409 views

Is there a (real) number which gives a rational number both when multiplied by $\pi$ and when multiplied by $e$?

Besides $0$ of course. What about addition and exponentiation? I would think there's no such number, but I'm not sure if I could prove it.
1
vote
1answer
275 views

Non-integer bases and irrationality

I read somewhere: When it comes to properties like prime, irrational, rational, divisible by 2, etc., nothing changes when you change base. But I'm not sure about the rational/irrational one. ...
5
votes
2answers
516 views

How to prove to be an irrational number? Like $\sqrt{2}$ $\sqrt{3}$ or $\sum\limits_{k=1}^{\infty} \frac{1}{n^2}=\pi^2/6$

As we know $\sqrt{2},\sqrt{3}$ are irrational numbers. And I see some proofs on the net. So I doubt that how $e,\pi$ or already known irrational numbers are proved to be irrational. In fact, I got ...
35
votes
2answers
955 views

Why is $\varphi$ called “the most irrational number”?

I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio ...
1
vote
1answer
126 views

Area of a circle is $A = \pi r^2$. Is it possible that both $A$ and $r$ are perfect integers.

Can you produce an example where both the area of a circle and it's radius are integers?
20
votes
1answer
528 views

Any proof to $\pi^{e}$'s irrationality?

I've searched for this for a while but get nothing... There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I ...
2
votes
1answer
118 views

Infinite irrational number sequences?

Is an irrational number, such as $\pi$ or $\sqrt2$, guaranteed to contain every possible digit sequence somewhere within it? Is there no proof for this? Is there any clue as to whether this is so? It ...
12
votes
2answers
240 views

Irrational numbers, decimal representation

Can this even be proved? (Or disproved?) Any irrational number without a 0 (zero) in its decimal representation is transcendental. Not sure where to start on this one...