0
votes
1answer
30 views

Calculating a floor sum

Is there any explicit closed form expression for $\sum_{k=1}^{\dfrac{p-1}2} \bigg\lfloor \dfrac{kq}p \bigg\rfloor-\bigg\lfloor \dfrac{k(q-1)}{(p-1)} \bigg\rfloor$ , where $p,q$ are odd primes ?
-4
votes
0answers
48 views

Identifying Symbols [on hold]

When you see $x$ written on a piece of paper you automatically identify it. When you yourself write $x^2 + 2x = 0$ The $x$ you write in $x^2$ differs from the $x$ you write in $2x$ just by a ...
-4
votes
0answers
65 views

Prove that $\sqrt{n}$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{n}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
1
vote
0answers
21 views

Find the reflection point $P$

On the real number line, paint red all points that correspond to points of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer points blue. Find a point $P$ on ...
3
votes
3answers
102 views

Solving $y^3=x^3+8x^2-6x+8$

Solve for the equation $y^3=x^3+8x^2-6x+8$ for positive integers x and y. My attempt- $$y^3=x^3+8x^2-6x+8$$ $$\implies y^3-x^3=8x^2-6x+8$$ $$\implies ...
2
votes
0answers
136 views

Closed formula for the numbers of the form $\sqrt{1+\sqrt{4+\sqrt{9}}}$

how can i find the formula for the nth term of this series? SQ = square root $\sqrt{1} = 1$ $\sqrt{1 +\sqrt{4}} = \sqrt{3}$ $\sqrt{1 +\sqrt{4+\sqrt{9}}} \approx 1.909385061$ $\sqrt{1 ...
0
votes
2answers
49 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
58 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
55 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
324 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
94 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
46 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
32 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
82 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
203 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
94 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
59 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
50 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
74 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
42 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
154 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
156 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
171 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
107 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
154 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
87 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
94 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
140 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
26 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
310 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
79 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
316 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
88 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
60 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
91 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 ...