1
vote
0answers
56 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
77 views

Finding set of non recurring non terminating decimals

I need to find a set of two Integers P and Q such that ...
9
votes
1answer
135 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
148 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
118 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
115 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?
12
votes
0answers
172 views

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

For $x\in\mathbb R$, let $\{x\} = x-\lfloor x\rfloor$ be the fractional part of $x$. I'm searching for an elementary proof that there exists a positive real number $s$ such that $$ \sum_{n = 1}^\infty ...
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
73 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
328 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
72 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
171 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
341 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
58 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 ...
-3
votes
1answer
210 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
174 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
267 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 ...
23
votes
2answers
428 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
82 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
137 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
150 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
136 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
76 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
78 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
330 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} = ...
84
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
396 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
214 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
479 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 ...
32
votes
2answers
757 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
116 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
502 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
114 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 ...
10
votes
2answers
180 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...
2
votes
1answer
130 views

Interesting question about irrational numbers

Find all solutions in un-ordered integers $(a,b)$ to $7-a-b=2\sqrt{10}-2\sqrt{ab}$. It would appear that the only solution to this is $a=2, b=5$. But how to prove this rigorously? Do irrational ...
4
votes
4answers
432 views

On comparing fractions , fraction with smaller difference between numerator and denominator is greater than the other

A text book proposed that "when comparing fractions ,if the compared fractions's are such that numerator is smaller than denominator ,then fraction with more difference(absolute) between numerator ...
0
votes
2answers
124 views

Is the Copeland–Erdős constant a random number? How is it normal?

The Champernowne constant is not random. Is the Copeland–Erdős constant random? Also if Copeland–Erdős number is normal, then shouldnt the number of $5$s and even digits be low because they cannot ...
1
vote
1answer
143 views

Type of periodicity in champernowne constant.

Digits of Champernowne constant are aperiodic, else it will be rational. Fine! But it is not random because I can write a program which will give me the position of every digit. E.g. I can calculate ...
17
votes
2answers
797 views

Does $\sin(x)=y$ have a solution in $\mathbb{Q}$ beside $x=y=0$

Is there a way to show, that the only solution of $$\sin(x)=y$$ is $x=y=0$ with $x,y\in \mathbb{Q}$. I am seaching a way to prove it with the things you learn in linear algebra and analysis 1+2 ...
3
votes
0answers
141 views

Must be rational number

Let $a$, $b$ positive rational number. Suppose that there exist two odd positive integers $p$, $q$ such that $\sqrt[p]{a}+\sqrt[q]{b}$ is rational. Prove that both $\sqrt[p]{a}$ and $\sqrt[q]{b}$ are ...
4
votes
1answer
136 views

Are these numbers $h_{r,s}$ irrational?

I came across these numbers in my work some time ago. This type of expressions do not exist in closed form (not to confuse with Vandermonde convolution), I already know that. To simplify I denote ...
20
votes
3answers
1k views

What is the simplest way to prove that the logarithm of any prime is irrational?

What is the simplest way to prove that the logarithm of any prime is irrational? I can get very close with a simple argument: if $p \ne q$ and $\frac{\log{p}}{\log{q}} = \frac{a}{b}$, then because ...
23
votes
1answer
2k views

Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
8
votes
5answers
363 views

Arithmetic of irrationals and the Vedanta behind it..

I am really curious about the Vedanta behind the arithmetic operations on irrational numbers. It still got aggrevated after the productive discussions with my friend. So I decided to ask it here. ...
2
votes
2answers
527 views

“GCD” of any two real numbers

This isn't really a GCD question, because GCD is only defined for integers. I'm interested in the the existence of a common divisor of any two non-zero real numbers. In other words can you prove or ...
5
votes
1answer
91 views

powers of $\frac{1+\sqrt a}2$

For any a which is not a perfect square, let $x=\frac{1+\sqrt a}2$. $x^n$ can be written uniquely as $b_nx+c_n$, where b and c are rational. Apart from $a=0, a=1, a= 1 \pm 2^m$ for $m>2$, are ...
4
votes
4answers
275 views

Is there an algorithm that can tell whether the power of two rational numbers is rational?

It has been known since Pythagoras that 2^(1/2) is irrational. It is also obvious that 4^(1/2) is rational. There is also a fun proof that even the power of two irrational numbers can be rational. ...
11
votes
3answers
568 views

Irrationality of “primes coded in binary”

For fun, I have been considering the number $$ \ell := \sum_{p} \frac{1}{2^p} $$ It is clear that the sum converges and hence $\ell$ is finite. $\ell$ also has the binary expansion $$ \ell = ...