0
votes
2answers
38 views

how shall i find the $n$-th term of this,

How shall I find the $n$-th term of this: $\sqrt{1+2}$ $\sqrt[3]{1+2+3}$ $\sqrt[4]{1+2+3+4}$ $\sqrt[5]{1+2+3+4+5}$ $\sqrt[6]{1+2+3+4+5+6}$ $\sqrt[7]{1+2+3+4+5+6+7}$ all the way to ...
2
votes
0answers
53 views

$\sum_{k=1}^n \lfloor kx \rfloor =$ ?

Let $x$ be a positive real number and $n$ a positive integer , then how may we evaluate $\sum_{k=1}^n \lfloor kx \rfloor $ ? If a closed form doesn't exist then can we at least find an asymptotic ...
3
votes
1answer
52 views

Nice polynomial reducibility: $x^n+4$

Problem: Find all $n\in \mathbb{N}$ such that $f(x)=x^n+4$ is reducible in $\mathbb{Z}[x]$. It seems $n=4k$ is the only one (the factorization follows easily from Sophie Germain's identity in this ...
1
vote
2answers
58 views

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3…10\}$?

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3...10\}$? I can't think of any way on how to generate such types of polynomials? Also, would they have a minimum ...
-1
votes
2answers
302 views

What is the efficient way to calculate number of divisors of N that are divisible by 2?. [closed]

For example if a number is given let say 8 then its factors are 1,2,4,8 hence total numbers of divisors which are divisible by 2 are (2,4,8) that is 3.
5
votes
2answers
87 views

Fermat's Last Theorem for Negative $n$

While studying Fermat's Last Theorem and Pythagorean triples, the following question occurred to me: For the equation $a^n+b^n=c^n$, where $n$ is a negative integer, a) does a solution exist, and b) ...
0
votes
0answers
21 views

How to find a certain uppper bound (see details)?

What would be the most efficient way to find this upper bound? Given natural number n and a natural number d < n, find the ...
3
votes
3answers
42 views

Finding domain of $f\text{ o }g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then ...
0
votes
1answer
30 views

Is there a quick way to obtain $a,b$ in $ax+by = z$ where $x,y,z$ are fixed and $x+1 = y$?

Suppose that all numbers are postive integers. Let $x,y,z$ be fixed/given and $x+1=y$. Then would there be a quick way to find set of solutions $(a,b)$ that satisfy $ax+by=z$? "Quick" would be ...
5
votes
3answers
78 views

$4^x+6^x=9^x$ $\implies$ $x \notin \mathbb Q$?

Does there exist any rational number $x$ such that $4^x+6^x=9^x$ ?
7
votes
2answers
197 views

$a-b,a^2-b^2,a^3-b^3…$ are integers $\implies$ $a,b$ are integers?

Let $a,b$ be distinct real numbers such that $a^n -b^n$ is integer for every positive integer $n$ , then is it true that $a,b$ are integers ?
3
votes
0answers
34 views

The n-th k-gonal number

I was doing some school work and got bored so I started messing with k-gonal numbers. I started with the triangular numbers, square numbers and looked for patterns. I noticed something. Let ...
3
votes
1answer
64 views

$(x^{2014}-x^{2004})\in \mathbb Z , \;(x^{2009}-x^{2004})\in \mathbb Z$ $\implies$ $x \in \mathbb Z$?

Let $x$ be a real number such that $(x^{2014}-x^{2004})$ and $(x^{2009}-x^{2004})$ are both integers. Then is $x$ also an integer ?
0
votes
2answers
58 views

Integer roots of quadratic equations

Does there exist real $b,c$ such that each of the equations $x^2+bx+c=0 \space,\space 2x^2+(b+1)x+c+1=0$ has two integer roots ?
2
votes
1answer
53 views

how to calculate all numbers of the form $6x-1, 6x+1, 6x+5$ that are not divisible by $5,7$ or $11$?

This is purely a hobbist question, I would simply like to know what methods are currently used to find the answer to this question. (Does modular arithmetic suffice in finding all the "$x$" values ...
0
votes
1answer
22 views

$n$ is some natural number. Let $x$ be the integer part of $\sqrt n$ and $y$ be the decimal part. If $x^2 - y^2 = 1+4y$ what is $y^x$?

$n$ is some natural number. Let $x$ be the integer part of $\sqrt n$ and $y$ be the decimal part. If $x^2 - y^2 = 1+4y$ what is $y^x$? This is some high school problem but I can't solve it. Any help? ...
0
votes
3answers
48 views

How to measure monotonicity of a list of values

I need to compare monotonicity of lists of values. I have $S=(n_1,n_2,...n_n)$, I need a function $\mathrm f(S)$ to return the monotonicity of the S. $S_1=[1,2,4,4,8]$ $S_2=[8,4,4,2,1]$ ...
0
votes
4answers
69 views

Prove that (integer + non-integer) never equals an integer.

My question is how do you prove that given an integer $x$ and a number $y$, the only way for $x + y$ to be an integer is if $y$ is also an integer. I can see how to prove by induction that an integer ...
0
votes
0answers
41 views

Module in $\mathbb{Z}$ and characteristics

I'd like to show some basic characteristics for a module in $\mathbb{Z}$. The module in Z has been defined as: "A subset $\not0 \not = M \subseteq \mathbb{Z}$ is called module, if M is closed under ...
1
vote
2answers
151 views

How to prove that certain integers and xy are solutions for a relation?

I am trying to solve the following problem: Let A be the set of all integers of the form a^2 + b^2 + 4ab where a and b are integers. Prove: a. if x and y are in A, prove xy is in A. b. Prove 121 is ...
3
votes
3answers
40 views

Divisibility in base $7$ problem

Find all integers between $0 \leq a \leq 2400$ such they are divisible by $8$ and that their base 7 development has at least $3$ equal digits.
2
votes
2answers
152 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. ...
2
votes
2answers
164 views

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I'm trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} ...
0
votes
1answer
100 views

ordered pair of unequal positive integer solution of $x+y+z+w = 20$

[1] Number of ordered pair of unequal positive integer solution of $x+y+z = 10$ [2] Number of ordered pair of unequal positive integer solution of $x+y+z+w = 20$ $\bf{My\; Try}::$ For $(1)$ one:: ...
0
votes
2answers
60 views

On finding the zeros of a polynomial

What is the zero (real) of the polynomial $$x^{k+1}-2x^{k}+1=0$$ If there is such, how can I find it or what method can I use?
1
vote
0answers
58 views

Show that $v$ mod $2$ = $w$ mod $2$ if and only if $\alpha$ mod $2$ = $\beta$ mod $2$

Basic setting: 1) Let $N$ be an even integer, and $a$ and $b$ are two odd integers in $\mathbb{Z}_N$ such that $\gcd{(N, a)}=\gcd{(N, b)}=1$. Thus there exist $a^{-1}$ and $b^{-1}$ in ...
3
votes
1answer
64 views

Maximum number of monic polynomials $f(x)$ such that for some real $a$, $x(f(x+a) - f(x)) = nf(x)$ for a fixed $n<1000$?

This was a problem that I did from a while ago that I am revisiting. I remember getting the answer as the maximum divisors of an $n < 1000$, which would be $n = 840 \rightarrow 32$ divisors and ...
2
votes
1answer
71 views

How to solve this equation without drawing it's graph

I'm curious to know how can I solve this equation without drawing it's graph . Can we ever prove that the answer must be integer using number theory or other mathematical methods? ...
3
votes
2answers
152 views

Proving $p\nmid \dbinom{p^rm}{p^r}$ where $p\nmid m$

A question from Advanced Modern Algebra by Joseph J.Rotman. Let $n=(p^r)m $ such that the prime $p\nmid m$.Prove that $p\nmid \dbinom{n}{p^r}$.HINT: Assume otherwise,cross multiply and apply ...
8
votes
4answers
250 views

Solve $3^x + 28^x=8^x+27^x$

The equation $3^x + 28^x=8^x+27^x$ has only the solutions $x=2$ and $x=0$? If yes, how to prove that these are the only ones?
1
vote
2answers
83 views

Is there an algebraic method to concat two numbers?

I'm searching an algebraic way to concat numbers in base $10$. Concatening two numbers is to put side by side their notations. Let $c$ a concatenating function. $c(2,2) = 22$ $c(8,9) = 89$ ...
5
votes
2answers
92 views

$P(P(\cdots(P(x))))$ and its integer solutions

Problem: Suppose that $P(x)$ is a polynomial with degree at least $2$ and integer coefficients. Let $Q(x)$ have the form $$ Q(x) = P(P(P(\cdots P(x) \cdots))) $$ for some finite number of nested $P$s. ...
1
vote
1answer
85 views

For any positive integer n, find the range of the product 1∗3∗5∗7∗9∗…∗(2n−1) in terms of n. [closed]

For any positive integer $n$, find the range of the product $1*3*5*7*9* \ldots *(2n-1)$ in terms of $n$. I have posted my answer...
14
votes
2answers
203 views

Prove $\log_5{30}<\log_8{81}$

It's easy to prove this by calculator or computer, and I wonder can we prove that $$\log_5{30}<\log_8{81}\tag 1$$ by pencil and paper ? Thanks in advance ! Edit: $(1)$ can be written as ...
0
votes
0answers
138 views

Counterexample for conjugate rules in $\mathbb{Z}[\sqrt[4]{2}]$

We all know that, in a field, such as $\mathbb{Z}[i]$ or $\mathbb{Z}[\sqrt[4]{2}]$, conjugate properties that as $ \overline {u \cdot v} = \overline {u} \cdot \overline {v} $ hold. Furthermore, ...
-1
votes
2answers
24 views

Show $ n + \operatorname{lcm}(a,b)\mathbb{Z} \subseteq (n + a \mathbb{Z}) \cap (n + b \mathbb{Z}) $

let $n \in \mathbb{Z}$. I want to show $$ n + \operatorname{lcm}(a,b)\mathbb{Z} \subseteq (n + a \mathbb{Z}) \cap (n + b \mathbb{Z}) $$ for integers $a,b$ TRY: Since $lcm(a,b)$ is multiple of ...
2
votes
1answer
298 views

Finding the remainder when a polynomial is divided by a product of numbers whose remainders are known

We have a polynomial $f(x)$ with rational roots that leaves remainders $15, 2x + 1$ when divided by the polynomials $x - 3, (x-1)^2$ respectively. What is the remainder when $f(x)$ is divided by $(x ...
3
votes
0answers
76 views

Solving $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$

If $m$ and $n$ are distinct positive integers then does the equation $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$ $\space$has any solution , for $x,y,z$ , in positive integers with $x,y,z$ all not equal ?
3
votes
1answer
297 views

Why n! equals sum of some expression?

Why n! equals sum of some expression? Especially I need to know why this expression is true? $$ n!= \left(\frac{n+1}{2}\right)^{p(n)} \; \prod_{j=0}^{q(n)}\sum_{i=0}^j(n-2i), $$ Where \begin{gather*} ...
2
votes
2answers
47 views

why is it constant?

$f$ is a function from $\Bbb R$ to $\Bbb R$: $\frac{f(x+y)}{(x+y)} - (x+y)^2 = \frac{f(x-y)}{(x-y)} - (x-y)^2$ for all $x$ and $y$ the solution book just says "Thus $\frac{f(x)}{x} - x^2$ is a ...
-1
votes
1answer
87 views

Simple Question about Induction?

let x be a natural number i want to prove that f(x)=$x^2$. suppose that f(x)=$x^2$, f(0)=0 holds we'll prove that f(x)= $(x+1)^2$, in the functional equation we have f(x-y)+f(x+y)=2f(x)+ stuff, ...
-1
votes
3answers
70 views

geometric sequence with product $P$

Given that the product of the first $n$ terms of a geometric sequence is $P$ , $ \bullet$ find $\displaystyle \prod_{k=1}^{\frac{n}{3}}a_{3k-1}$ in terms $P$ if the terms are $a_1,a_2,a_3...a_n$ $ ...
1
vote
2answers
51 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
89 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
181 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
49 views

Is there a way to change the order of these summation terms

$$\sum_{K=2}^{N}\sum_{L=1}^{\lfloor\frac{K}{2}\rfloor-1}$$ I want to have the $L$ summation on the outside and the $K$ summation on the inside somehow. Can this be done?
2
votes
3answers
246 views

Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ [duplicate]

How to show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ for coprime $a$ and $b$? I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and $\gcd(a^2,b)=1$, but how do I apply this to that?
1
vote
1answer
256 views

Is $\large \frac {\pi}{e}$ rational, irrational, or trandescendal?

Is there an argument for why $\large \frac {\pi}{e}$ is rational, irrational, or trandescendal? Can the quotient of any two transcendental numbers (which are not rational multiples of each other) be ...
2
votes
4answers
117 views

Confusion with Factor $(2n)!$

Good Day folks ... Next, I'm here doing an exercise divisibility, and the next, I came across something that confused me $$(2n)!=?$$So, how is it?$$2!n!;?$$$$2(n!);?$$thinking ...
2
votes
1answer
127 views

Mill's formula, $\theta^{3^n}$ is a prime for a certain $\theta$ and all natural $n$?

I just watched this video done by numberphile, and the video claimed that there exist certain numbers $\theta$ such that the floor function of $\theta^{3^n}$ is a prime for all natural $n$ . $\theta$ ...