Tagged Questions
0
votes
0answers
41 views
Matching numbers by $f(x)=\frac{1}{x}$
Let $0<x \leq 1$, We define a function such that $f(x)=y=\frac{1}{x}$ which results $y \geq 1$ . We have infinitely many numbers between $0$ and $1$, so we can match any $x$ to a number $y$ greater ...
3
votes
4answers
112 views
Find $a,b,c \in \mathbb {Q}$
Find $a,b,c \in \mathbb {Q}$ such that:
$\left\{\begin{array}{rl} x^3&\in \mathbb Q \\ x&\notin \mathbb{Q}\\ ax^2+bx+c &=0\end{array}\right.$
I tried Vieta's formulas, but seem like it ...
0
votes
1answer
39 views
Easy way to check for a valid solution in this triple equality?
Let's say I have the following equalities
$a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 = b_1x_1 + b_2x_2 + b_3x_3 + b_4x_4 = c_1x_1 + c_2x_2 + c_3x_3 + c_4x_4$
Where the $a$'s, $b$'s, and $c$'s are known, ...
2
votes
2answers
62 views
Find the minimum values of $a,b,c,d,e,f$ that satisfy following equations
${ a }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c \right) }^{ 2 }={ d }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c+d \right) }^{ 2 }={ e }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c+d+e \right) }^{ 2 }={ ...
1
vote
1answer
41 views
A Diophantine equation and decimal digits
Solutions of the Diophantine equation
$a10^n+(a+1) = (2^{m+1}-1)*2^{m+1}$
are
12=3*4,
56=7*8,
67100672=8191*8192.
Are there more solutions/examples like that or a generalization of the ...
2
votes
0answers
36 views
Unique decomposition of $c$ sums of products of $k$ numbers greater than 1, allowing duplicates?
This question differs from Unique decomposition of $c$ sums of products of $k$ prime numbers, allowing duplicates? in that prime number restriction is changed to any number greater than 1.
Suppose ...
2
votes
0answers
40 views
Unique decomposition of $c$ sums of products of $k$ prime numbers, allowing duplicates?
Suppose that there are $n$ different prime numbers. Define procedure a) as following ($k \leq n$ and $k$ fixed): procedure a) for each time, we select one number out of $n$ possible cases and multiply ...
1
vote
4answers
70 views
I cannot find the last factor of this expression?
I'm supposed to factor $x^8-y^8$ (the exponents are 8 for both if it is too difficult to see) as completely as possible. It is easy to factor this to $(x+y)(x-y)(x^2+y^2)(x^4+y^4)$. However, the book ...
27
votes
6answers
568 views
Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?
In this recent answer to this question by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$
...
1
vote
1answer
96 views
Alternative solutions to $n^2+n = k^2+k + 2kn$
Consider this equation:
$n^2+n = k^2+k + 2kn$
I want to find the set of non-negative integer n,k that satisfies the equation.
I tried to write $n$ as $k$ by solving the equation with $n$ as root ...
0
votes
3answers
63 views
Finding the remainder
How to find the remainder for :
$$x^{81}+x^{49}+x^{25}+x^{9}+x$$
divided by :
$$x^3-x$$
1
vote
1answer
53 views
How to find $a,b\in\mathbb{N}$ such that $c = \frac{(a+b)(a+b+1)}{2} + b$ for a given $c\in\mathbb{N}$
Suppsoe that $$c = \frac{(a+b)(a+b+1)}{2} + b$$
Now $c$ is given - how does one find satisfying $a, b$?
0
votes
2answers
17 views
For what range does this floor function scale to?
I have $\lfloor\frac{X}{(2y+1)^2}\rfloor = k$ where $X$ and $k$ are known. For what values of $y$ will this hold true?
edit: all are positive integers
1
vote
1answer
66 views
$n$ fractional gaussian rational numbers and set of rules to govern them
Suppose that there are $n$ Gaussian rational numbers of form $a_t + b_ti$. For any $t$th number, $a_t$ and $b_t$ are both constrained to be non-zero rational numbers (fractions). Suppose that we ...
2
votes
0answers
112 views
Imposing condition of specification of product of $n$ of imaginary numbers on coefficients of imaginary numbers
I asked the same question but with some fatal mistake that makes the question unanswerable - so I decided to delete it and start new.
Connecting from The set of numbers that when multiplied do not ...
0
votes
1answer
42 views
The set of numbers that when multiplied do not get decomposed into $sx+ty$ while the numbers themselves are of form $ax+byi$
Suppose that there are $n$ numbers that are in the following format: $a+bi$. Each number has different combination of $a$ and $b$. $a,b$ must be non-zero integers.
Suppose that we impose the ...
1
vote
1answer
33 views
Condition so that an integer $x$ is $x \neq pa + qb$
Suppose that I want to form an integer $x$ so that $x \neq pa + qb$ where $a$ and $b$ are fixed integers(that is, these integers are set from the start) and $p,q$ are free integers. All numbers are ...
1
vote
3answers
41 views
Finding an easy way to find decomposition nonzero integer $x$ into $x = sa + tb$?
Suppose that some nonzero integer $x$ is $x = pa + qb$ where $p, q, a, b$ are also nonzaero integers.
What would be the easy way to find another decomposition of $x$ into following: $x = ra + sb$ ...
1
vote
3answers
107 views
Finding the integer solutions
Find all integer solutions of
$$(a + b^2)(a^2 + b) = (a − b)^3.$$
Obviously $b = 0$ is one. But how to get other solutions?
0
votes
1answer
128 views
Finding set of values that satisfy constraint equation
We want to find set of all values that satisfy the following equation: $(a+ky)(a-ky)=gx$ All values are assumed to be nonzero integers. How does one set $x$ so that $a$ is not multiples of $y$ while ...
2
votes
1answer
74 views
Is there a way to rewrite this without the ceil() function?
$$L = \left\lceil \frac{\sqrt{v-4 \times N}-1}{4} \right\rceil$$
This is a line in my program but I cannot get ceil() to work in GMP, so I'd like to approach this mathematically and just rewrite it ...
0
votes
1answer
50 views
Converting loop to a closed form expression? [duplicate]
Possible Duplicate:
How to convert this loop into a closed form expression?
I have the following code in Python
...
2
votes
1answer
45 views
Finding an equation of solutions of an equation given constraints
$$\left(\frac{ax}{p_1}\right)^2 - \left(\frac{bx}{p_2}\right)^2 = cx \quad\text{and}\quad a\neq p_1, \, \, b \neq p_2$$where $a,b,c$ are nonzero integers, and $p_1$ and $p_2$ are distinct prime ...
9
votes
2answers
166 views
Determining the number $N$
Let $1 = d_1 < d_2 <\cdots< d_k = N$ be all the divisors of $N$ arranged in increasing order. Given that $N=d_1^2+d_2^2+d_3^2+d_4^2$, determine $N$. The divisors include $N$. It seems that ...
9
votes
3answers
151 views
numbers' pattern
It is known that
$$\begin{array}{ccc}1+2&=&3 \\ 4+5+6 &=& 7+8 \\
9+10+11+12 &=& 13+14+15 \\\
16+17+18+19+20 &=& 21+22+23+24 \\\
25+26+27+28+29+30 &=& ...
7
votes
1answer
98 views
“reverse” diophantine equation
Suppose we define $A= (8+\sqrt{x})^{1/3} + (8-\sqrt{x})^{1/3}$. How can we find, algebraically, all values of x for which $A$ is an integer?
I was not able this problem save for with Mathematica. How ...
0
votes
1answer
55 views
How to find out the probability of ordered pairs of rational or irrational number $(a,b)$ such that $1<a<50, 1,<b<50$, and $\log_b a$ is rational.
How to find out the probability of ordered pairs of rational or irrational or transcendental number $(a,b)$
such that $1<a<50$, $1<b<50$, and $\log_b a$ is rational?
Uniformly ...
5
votes
2answers
179 views
An Interesting Question about Pythagorean Triples
I have recently thought about a interesting question about Pythagorean Triples.
Consider such a right-angled trapezium formed by 3 right-angled triangle. Determine does it exist integral ...
5
votes
5answers
326 views
How to disprove there exists a real number $x$ with $x^2 < x < x^3$
I realize that the only method is to show various cases:
I must test for $x > 1$, $x < -1$, $0 \leq x \leq 1$, and $-1\leq x \leq0$.
But even with this, I don't understand how to inject the ...
2
votes
3answers
164 views
How to prove floor identities?
I'm trying to prove rigorously the following:
$\lfloor x/a/b \rfloor$ = $\lfloor \lfloor x/a \rfloor /b \rfloor$ for $a,b>1$
So far I haven't gotten far. It's enough to prove this instead
...
2
votes
5answers
214 views
If $p, q$, and $r$ are relatively primes, then there exist integers $x$, $y$, and $z$ such that $px + qy + rz = 1$
True/False
If $p, q$, and $r$ are relatively primes, then there exist integers $x, y$, and $z$ such that $px + qy + rz = 1$
NOTE: $p, q$, and $r$ are positive primes.
4
votes
3answers
187 views
Largest number that divides $n^2(n^2 - 1)(n^2 - n - 2)$ for all $n$
Obtain the greatest natural that divides $n^2(n^2 - 1)(n^2 - n - 2)$ for all natural numbers $n$.
What should be the approach in these type of questions?
Should I equate with prime factorization ...
2
votes
2answers
91 views
How to obtain the number of digits in n!?
How to obtain the number of digits in $n!$ ?
My approach :
I Used Stirling's formula to find out the approximate value of $n!$
Let the approximate value be $S$
Thus, number of digits in $\ = ...
4
votes
5answers
146 views
Find the possible value from the following.
Find the possible value from the following.
I'm not able to end up on a concrete note, as I'm unable to get the essence of question, still not clear to me.
$x$, $y$, $z$ are distinct reals such ...
0
votes
1answer
61 views
what is general theory for those type of problem: To think about the condition must be exist if a decimal with infinite digit in base 10 …
what is general theory for those type of problem: To think about the condition must be exist if a decimal with infinite digit in base 10 also have infinite digit in both base 3, base 4? please prove ...
2
votes
1answer
291 views
Sum of GCD(k,n)
I want to find this
$$ \sum_{k=1}^n \gcd(k,n)$$
but I don't know how to solve. Does anybody can help me to finding this problem.
Thanks.
0
votes
2answers
81 views
What is a general way to find out whether a number obtained by a finite combination of algebraic operations is algebraic?
What is a general way to get a integers inside a radical with + or - operation(the numbers adding or subtracting each others, for example, $\sqrt5 +\sqrt7$ is this type of numbers)allow is algebraic ...
15
votes
4answers
633 views
Proving that $\sum\limits_{i=1}^k i! \ne n^2$ for any $n$ [duplicate]
Possible Duplicate:
How to prove that the number 1!+2!+3!+…+n! is never square?
Show that $\displaystyle\sum\limits_{i=1}^k i!$ is never a perfect square for $k\ge4$
I could prove ...
1
vote
0answers
146 views
variants of geometric series
My question can $\displaystyle \mathbf{\sum_{n \geq 0} a^{\lfloor n \sqrt{2}\rfloor}}$ be expressed as the sum of rational functions in a? Here $\lfloor \alpha \rfloor$ is the floor function, the ...
0
votes
2answers
387 views
Factoring Multiple Variable Polynomials
This is in relation to a problem dealing with the three-dimensional analogue of Pell's Equation. I would like to factor
$ x^3+Dy^3+D^2z^3-3Dxyz $ into $\frac{1}{2}(x+Dy+D^2y)$ and another factor.
I ...
1
vote
4answers
250 views
If x=1, then its powers end with 1
How to "show", that if x=1, then its powers will end with 1, if x=5, it's powers end with 5.
10
votes
4answers
574 views
Not quite Fermat's Last Theorem
Prove that the equation na + nb = nc, with a,b,c,n positive integers, has infinite solutions if n=2, and no solution if n≥3.
