Tagged Questions
12
votes
3answers
301 views
$\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number
I need help to prove the following result.
$\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.
3
votes
1answer
61 views
Minimum number of coconuts
Three friends namely $A$, $B$ and $C$ collected coconuts with the help of monkey and fell asleep. At night, $A$ woke up and decided to have his share. He divided coconuts into three shares, gave the ...
6
votes
1answer
134 views
All number fields with absolute value of discriminant $\le 20$
I need to find all number fields with absolute value of discriminant $\le 20$.
Using Minkovsky's theorem I understood that it should be quadric or cubic extension. The case of quadric is very ...
2
votes
2answers
110 views
Last 2 numbers of the product of divisors
Let $N$ be the product of all divisors of $2013^{2013}$. What are the
last 2 numbers of $N$ in its decimal notation?
I don't know where to start in this exercise, would like to get a hint or ...
6
votes
2answers
41 views
Number of elements of a matrix subset with field $\mathbb{Z}_p$
Can anybody please help me this problem?
Let $K = \mathbb{F}_p$ be the field of integers module an odd prime $p$, and $G = \mathcal{M}^*_n(\mathbb{F}_p)$ the set of $n\times n$ invertible matrices ...
1
vote
2answers
43 views
Solution of equations
Consider the following system of equivalences of integers.
$$x \equiv 2 ~~\text{mod} ~~ 15$$
$$x \equiv 4 ~~\text{mod} ~~ 21$$
We need to find the number of solutions in $x$, where $1 \lt x \lt 315$, ...
2
votes
2answers
27 views
Show that ord$_{p}2 = 2^{n + 1}$.
Let $p$ be a prime divisor of the Fermat number $F_{n} = 2^{2^{n}} + 1$.
Show that ord$_{p}2 = 2^{n + 1}$
The order of the element modulo some integer is the least positive integer such that it ...
0
votes
0answers
32 views
$\langle v,\sqrt{2}v\rangle_{\mathbb{Z}}$ not a discrete subgroup of $\mathbb{R}^{2}$ [duplicate]
I got a list of exercises to do and there is one of the first exercises which I do not manage to solve.
Its statement is the following:
Let $v\in \mathbb{R}^{n}$ be a nonzero vector. Using the fact ...
0
votes
1answer
46 views
Ring Isomorphism Proof
Let $p$ be a prime with $p \equiv 1 (\mod 4 )$.
I am trying to show that $\mathbb{Z}[X]/(X^2 + 1, p) \cong \mathbb{Z}_p \times \mathbb{Z}_p$ is a ring isomorphism.
I am not really sure how to ...
0
votes
2answers
19 views
Binary Vector Communication
Alice holds an $n$ x $n$ binary matrix $A$, and Bob holds an $n$ x $n$ binary matrix $B$. They want to check whether $A = B$, but they do not want to communicate too much. Here is what they do: Alice ...
1
vote
1answer
40 views
How to concisely prove that if a is bigger than one, then $(a \mid (b \mod a+c \mod a)) \equiv a \mid b+c$.
Show that if $a \in \mathbb{Z_{\geq 0}}$, then the following proposition holds:
$a \mid [b {\pmod{a}}+c \pmod {a}] \iff a \mid b+c$.
I've been trying to prove it, but I am blocked. I tried using ...
0
votes
1answer
32 views
Show that $(p^{n+1} - 1)/(p -1)$ is even if and only if $n+1$ is even
Show that if $p \in \mathbb{Z}_+$ is odd and $\geq 3$, and $n \in \mathbb{Z}_+$, then $(p^{n+1} - 1)/(p -1)$ is even if and only if $n+1$ is even, and $(p^{n+1} - 1)/(p-1) \equiv 0 \mod 4$ if and ...
1
vote
1answer
33 views
Sum of the Hyperharmonic\Over-harmonic Series under $\mathbb{Z}_n$ for $p=2$
For $n \geq 5$ prime number, calculate the sum of:
$$1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{(n-1)^2}$$
under $\mathbb{Z}_n$.
I figured it's the hyperharmonic\over-harmonic series,
$$ ...
1
vote
2answers
71 views
Show that there are infinitely many primes which are $\pm 1 \mod 5$
Show that there are infinitely many primes which are $\pm 1 \mod 5$.
HINT: Suppose that there are finitely many such primes, and let these primes be $q_i$ for $1 \leq i \leq n$. Let
$$N = ...
0
votes
1answer
47 views
Find if a number $n$ is a primitive root of $p$
Let $n = p_1\cdot p_2\cdot\ldots\cdot p_k$ where the $p_i$ are primes. Let $s = \varphi(n)$ where $\varphi$ denotes the Euler Totient Function.
If none of $p_1,p_2,\ldots,p_k$ makes $a^{(s/p_i)} = 1$ ...
-1
votes
1answer
54 views
Number of partitions of n with k parts.
Prove the following identities:
a) $p_k (n) = p_k (n-1) + p_k (n-k)$
b)$p_k (n) = \sum\limits_{s=1}^{k} p_s (n-k)$
where $p_k (n)$ denotes the number of partitions of $n$ with $k$ parts.
1
vote
2answers
37 views
Find the values of $p$ such that $\left( \frac{7}{p} \right )= 1$ (Legendre Symbol)
Show that if $p$ is an odd prime coprime to $7$, then $\left( \frac{7}{p} \right) = 1$ if and only if $p \equiv \pm 1, \pm 3,$ or $\pm 9 \pmod{28}$. HINT: If $p$ is an odd prime, determine which ...
1
vote
0answers
28 views
3SAT to Subset Sum
I am starting with a 3-CNF $F(x_1,...,x_n) = C_1 \wedge ... \wedge C_m$.
First, ow do I show that there is no clause that contains both a variable and its negation?
Then, I have the set of equations ...
3
votes
0answers
42 views
Use the ring ${\bf{O}}[\sqrt5]$ to show that $2$ is irreducible in $\mathbb{Z}[\sqrt5]$
I have this question:
Define
$${\bf{O}}[\sqrt5] = \{c_1 + c_2 \sqrt5 : c_1 + c_2 \in \mathbb{Z} \wedge c_1 - c_2 \in \mathbb{Z}\}.$$
This ring properly contains $\mathbb{Z}[\sqrt5]$. The ...
0
votes
3answers
54 views
Modulo number multiplied by constant
I am proving that for any integers $a,b$, it is impossible to write $a^2 - 5b^2 \equiv 2 \mod 4$. The first thing I have said is to assume $a,b$ are both even. So I have said
$$a,b \equiv 0 \, \, ...
0
votes
2answers
43 views
Inverse bit in Chinese Remainder Theorem
I need to solve the system of equations:
$$x \equiv 13 \mod 11$$
$$3x \equiv 12 \mod 10$$
$$2x \equiv 10 \mod 6.$$
So I have reduced this to
$$x \equiv 2 \mod 11$$
$$x \equiv 4 \mod 10$$
$$x ...
1
vote
1answer
26 views
Groups $G$ of order $8$ so that $U(\mathbb{Z}/n\mathbb{Z})\cong G$ for some $n$.
I cant solve this exercise.
Find all groups $G$ of order $8$ so that $U(\mathbb{Z}/n\mathbb{Z})\cong G$ for some $n$.
I need a little help here. thanks!!!
1
vote
1answer
62 views
prove that $U_{51}$,$U_{80}$ are not isomorphic
I need prove the next result:
$U(\mathbb{Z}/51\mathbb{Z})$,$U(\mathbb{Z}/80\mathbb{Z})$ are not isomorphic.
thanks for your help!
1
vote
1answer
48 views
Groups of units: Find an explicit isomorphism $U_{35}$, $U_{39}$
I need help in the following exercise:
Find an explicit isomorphism between $U(\mathbb{Z}/35\mathbb{Z})$ and $U(\mathbb{Z}/39\mathbb{Z})$.
Thanks!
2
votes
4answers
91 views
Show that if $c_1 + c_2\sqrt{5}$ divides $n$ in ${\bf{O}}[\sqrt{5}]$, then so does $c_1 - c_2\sqrt{5}$
I have a ring:
$${\bf{O}}[\sqrt{5}] = \{c_1 + c_2\sqrt{5}: (c_1 \in \mathbb{Z} \wedge c_2 \in \mathbb{Z}) \lor (c_1 + \frac{1}{2} \in \mathbb{Z} \wedge c_2 + \frac{1}{2} \in \mathbb{Z}) \}.$$
I ...
4
votes
2answers
51 views
How to factorise a polynomial in the ring $\mathbb{Z}_p[x]$
I previously asked a question: How is $x^2 + x + 1$ reducible in $\mathbb{Z}_3[x]$?, and I got told there that you could factorise $x^2 + x + 1 = (x + 2)(x+2) = (x-1)(x-1)$ in the ring ...
1
vote
1answer
33 views
$H_1, H_2$ groups. $|H_1 |= n_1, |H_2| = n_2$. Prove $|H_1 \times H_2| = n$ where $n = \mathrm{lcm}(n_1,n_2)$
Let $H_1$ and $H_2$ be groups. Prove that if $a_1 \in H_1$ has order $n_1$ and $a_2 \in H_2$ has order $n_2$, then the order of $(a_1, a_2)$ in $H_1 \times H_2$ is $n$, where $n = \mathrm{lcm}(n_1, ...
4
votes
1answer
49 views
For $1 \leq q < p \leq 8$, identify which $p^2 + q^2$ is not prime (Pythagorean Triples)
In my previous question: Calculating Pythagorean triples, I used
$$x + iy = (p + qi)^2 \hspace{1cm} z = p^2 + q^2$$
to calculate a load of Pythagorean triples (of the form $x^2 + y^2 = z^2$) where ...
2
votes
1answer
46 views
Calculating Pythagorean triples
If $x$ and $y$ are even, then of course $z$ is too, and $\left(\frac{x}{2}, \frac{y}{2}, \frac{z}{2} \right)$ is also a Pythagorean triple. For this question we assume that $(x, y, z)$ is a ...
4
votes
1answer
57 views
how compute $\gcd\Bigl(\binom{n}{1},\binom{n}{2},\binom{n}{3},…,\binom{n}{n-1}\Bigr)?$
how to prove $$\gcd\Biggl(\binom{n}{1},\binom{n}{2},\binom{n}{3},...,\binom{n}{n-1}\Biggr)=\begin{cases}
p, & \text{$n=p^m$ ;$p$ is prime} \\
1, & \text{o.w} \\
\end{cases}$$
thanks in ...
2
votes
3answers
74 views
GCD of polynomials over GF(p)
I have two polynomials:
$$
f(x)=(x^2+1)(x-2)
$$
$$
g(x)=(x^3+7)(x-2)
$$
I am supposed to find their GCD over GF(p) for some prime p.
I "understand" that their GCD is $(x-2)$ but what is the meaning ...
0
votes
2answers
109 views
How to calculate: $\displaystyle \frac{1}{2}+\frac{3}{4}+\frac{5}{6}+…+\frac{99}{100} $
How can I calculate value of $\displaystyle \frac{1}{2}+\frac{3}{4}+\frac{5}{6}+.....+\frac{97}{98}+\frac{99}{100}$.
My try:: We Can write it as $\displaystyle \sum_{r=1}^{100}\frac{2r-1}{2r} = ...
1
vote
1answer
68 views
Coding Theory - Fourier/Walsh/Hadamard Transform
Hi guys these questions are from my homework. I am not asking you to solve my homework. Instead, i just need some help in getting started. Please list down some steps as I am very lost. Also, if you ...
4
votes
1answer
240 views
Partial sum of numbers
My TA gave today this question as a nice question to think about. He said its involves standard ideas of Probability theory and numbers. But, I don't even know how to start.
Let $x_1, \ldots, x_n$ ...
4
votes
1answer
133 views
Solving $x^2+19=y^5$
I was given several exercises and there is a particular one, I am not able to solve.
Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...
1
vote
1answer
48 views
3 complex-variable equation
Moderator Note: This is a current contest question on Brilliant.org.
$x,y,z$ are complex numbers satisfying
$$
\begin{align}
x+y+z & =1\\
x^2+y^2+z^2 & =2\\
x^3+y^3+z^3 & =3
...
1
vote
2answers
79 views
Ordered triples solution to system of equations
How many ordered triples $(x,y,z)$ of integer solutions are there to the following system of equations?
$$
\begin{align}
x^2+y^2+z^2&=194 \\
x^2z^2+y^2z^2&=4225
\end{align}
$$
2
votes
1answer
45 views
Calculate or bound infimum
Let $a_1, \ldots, a_n \in\mathbb R$ and nonnegative let $b\geq1$ and $c\in [0,1]$.
Calculate or bound from above
$$
\inf \left\{d>0: \sum_{i=1}^n \ln ...
0
votes
0answers
28 views
Determining a Decidable Language
I am trying to decide which of the following problems are decidable:
a) To determine, given a Turing machine $M$ and a state $q$, where there is any configuration at all which yields a configuration ...
0
votes
4answers
47 views
Prove $2^k+1$ divisible by 3 for odd K
Prove $2^k+1$ divisible by 3 iff $k$ is odd number.
Since I need to prove both direction looks like if I need to prove it's divisible by 3 it's by induction and the other side by congruence..am I ...
3
votes
1answer
98 views
Sum of number of divisors
What is the value of d(1) + d(2) + d(3) + ... + d(99)?
Here d(x) denotes the number of positive divisors of x including 1 and x.
For example, positive divisors of 4 are 1, 2, and 4 so d(4)= 3.
4
votes
1answer
119 views
Dirichlet L-series and Gamma function question
Could someone help me, please, with this exercise?
Consider a sequence of complex numbers $\{a_n\}$ such that $a_n=a_m $ iff $ n\cong m $ mod $q$ for some positive integer $q$.
Define the ...
6
votes
1answer
71 views
How to prove $\sum_{i=1}^{n-1}\frac{1}{\operatorname{lcm}(a_i,a_{i+1})}\lt1$ where $a_i\in\mathbb N$ and $a_i\lt a_{i+1}$?
Let $a_1,a_2,\ldots ,a_n\in\mathbb N$ and $a_1\lt a_2\lt\cdots\lt a_n$. Then how to prove $$\sum_{i=1}^{n-1}\frac{1}{\operatorname{lcm}(a_i,a_{i+1})}\lt1$$
Thanks in advance
1
vote
2answers
68 views
Sum of divisors is prime implies number of divisors is prime.
I've seen this posted but I haven't seen this in depth as i need it. I turned this in as homework but only got 1 out of 3 on it, so any clarification would be wonderful.
Show that if the sum of all ...
1
vote
2answers
74 views
Number Theory - Primitive roots
Let $n$ be positive integer.
$G$ is cyclic and finite group generated by $g$.
I need to prove that for every positive integer $k$ :
$$O(g^k) = \frac{n}{\gcd(n,k)}$$
I don't really know what to do ...
2
votes
1answer
44 views
Injection? $f(x)=3x-\ \mid x \mid + \mid x-2 \mid$
For the function how do I tell if it's injective or not? There's two absolute value signs and is confusing me!
$$f(x)=3x\ - |x| + |x-2|$$
Thanks for the help!
0
votes
2answers
44 views
Proving two Complexes' Numbers Properties
I'm having problem working with complex number on this question and was wondering if someone can walk through with me their reasoning on how to solve this/these types of questions. Thanks in advance!
...
2
votes
2answers
104 views
Factorization of ideals in $\mathbb{Z}[\sqrt{5}]$
Consider the ring $R=\mathbb{Z}[\sqrt{5}]$. Let $I$ be the following ideal of $R$:
$$I:=(3,1+\sqrt{5})$$
My teacher said that the following equation holds:
$$I^2=(3)I,$$
but I actually can't ...
0
votes
1answer
223 views
Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$?
Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$?
$Approach$:
$N$=$11^2$.$13^4$.$17^6$
$N^2$=$11^4$.$13^8$.$17^{12}$
This ...
4
votes
1answer
299 views
Find the number of pairwise coprime triples of positive integers (a,b,c) with a<b<c
Find the number of pairwise coprime triples of positive integers $a,b,c$ with $a\lt b\lt c$ such that
a|bc−31, b|ca−31, c|ab−31
Details ...
