Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.
2
votes
0answers
21 views
Career in Number Theory?
I am about to get my B.S. in Mathematics, and I will be applying for PhD in pure mathematics next year, with future plans of teaching and doing research. Over the past year, I have developed a great ...
3
votes
4answers
62 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
0answers
34 views
Amount of Background Needed for Number Theory Research
How much background is needed to do research pure number theory? I mean things like descriptions under 18.785 and 18.786 in http://student.mit.edu/catalog/m18b.html. I get the impression that it takes ...
4
votes
2answers
51 views
Number Theory $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd
If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.
Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with
...
3
votes
0answers
36 views
Multiplicatives [duplicate]
Let $f: N \to N$, $f(2) = 3$, and $f(ab) = f(a)f(b)$, that is, f is a multiplicative function. f is also strictly increasing. Show that no such function exists.
Progress: Apparently, this is proven ...
3
votes
0answers
31 views
System of Diophatine equations $x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ y \in \mathbb{Z^+}$
Let $a_1,\ a_2,\ a_3,\ \ldots,\ a_n$ be distinct positive integers.
Find $x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ y \in \mathbb{Z^+}$
such that: $$\left\{\begin{array}{rl}(x_1,x_2,\ldots,x_n)&=1\\ ...
2
votes
0answers
42 views
finding out linear decomposition of $x$ into $k$ prime numbers
Some $k$ prime numbers $n_1, n_2, ..., n_k$ are given. Then some natural number $x$ is provided.
Then we want to figure natural numbers (including zero) $m_1, m_2, ..., m_k$ so that $n_1m_1 + n_2m_2 ...
0
votes
1answer
49 views
Probability of two random n-digit numbers dividing each other
Let $n$ be a positive integer. Suppose $a$ and $b$ are randomly (and independently) chosen two $n$-digit positive integers which consist of digits 1, 2, 3, ..., 9. (So in particular neither $a$ nor ...
11
votes
0answers
120 views
What is the smallest integer $n$>1 such that $n^{5000}+n^{2013}+1$ is prime?
Which is the smallest integer $n>1$, such that $$n^{5000}+n^{2013}+1$$ is prime ?
Since $x^{5000}+x^{2013}+1$ is irreducible over $\mathbb{Q}$ and has value $1$ for $x=0$,
there should be ...
3
votes
0answers
48 views
how prove $\phi(n)\ge \frac{n}{6\log \log (n)} $ $\forall n\ge5 $
How to prove$\forall n\ge5 $
$$\phi(n)\ge \frac{n}{6\log \log (n)} $$
$\phi$ is Euler function
Thanks in advance
0
votes
1answer
39 views
Easy way to check for a valid solution in this triple equality?
Let's say I have the following equalities
$a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 = b_1x_1 + b_2x_2 + b_3x_3 + b_4x_4 = c_1x_1 + c_2x_2 + c_3x_3 + c_4x_4$
Where the $a$'s, $b$'s, and $c$'s are known, ...
2
votes
1answer
96 views
convergence of the sum of reciprocals of “fake twin” primes
This question is inspired by the announcement of a proof that "fake twin" primes, i.e. pairs of consecutive primes differing by at most K, are -in infinite number- where K is a fixed integer which can ...
9
votes
2answers
75 views
Showing that a real number is an algebraic integer
For what values of $x,y,z\in\mathbb{Z}$, such that $0\leq x,y,z\leq 2, $ the real number $$\alpha:=\frac{1}{3}\left(x+\sqrt[3]{175} \cdot y+\sqrt[3]{245}\cdot z\right)$$ is an algebraic integer i.e. ...
4
votes
3answers
75 views
How do you prove that the mean of the co-primes of a number is half the number?
Say $n = 6$, The set of co-primes is $\{1, 5\}$, $\text{mean} = 3$
For $n = 9$, the set of co-primes is $\{1, 2, 4, 5, 7, 8 \}, \text{mean} = 4.5$
Question: Prove that the mean of co-primes of ...
2
votes
2answers
60 views
Find the minimum values of $a,b,c,d,e,f$ that satisfy following equations
${ a }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c \right) }^{ 2 }={ d }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c+d \right) }^{ 2 }={ e }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c+d+e \right) }^{ 2 }={ ...
12
votes
3answers
294 views
$\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number
I need help to prove the following result.
$\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.
3
votes
3answers
64 views
Right triangles with integer sides
Most of you know these triples:
$3: 4 :5$
$5: 12 :13$
$8: 15 :17$
$7: 24 :25$
$9: 40 :41$
More generally we can construct such triangles such as
$$2x:x^2-1:x^2+1$$
My question is why one of ...
4
votes
0answers
81 views
The divergence of the series of reciprocals of primes (proof check):
I just wanted to check my attempt at a proof for the divergence of:
$$\sum_{n=1}^{\infty} \frac{1}{p_n} \tag{ $\star$ }$$
We begin with assuming that $(\star)$ converges. If $(\star)$ ...
1
vote
1answer
39 views
Elements and their properties in a finite field
I need help proving the following.
If $\alpha \in (\mathbb{Z}/p\mathbb{Z})[x]/\langle f\rangle$ for some irreducible $f\in (\mathbb{Z}/p\mathbb{Z})[x]$ of degree $n$, then both ...
4
votes
2answers
34 views
Linear polynomials of finite fields
I have a final tomorrow, and I was looking over some exercises in my textbook. However, I can't seem to work this problem out.
Let $F$ be a field of $p^n$ elements and let $\alpha \in F^*$, where ...
4
votes
0answers
42 views
A numerical coincidence with continued fractions
My brother built a garage that measures 45 feet by 30 feet. To make sure the right angles were accurate, he measured the two diagonals of the rectangle to see that they were equal. In inches,
$$
...
-2
votes
1answer
121 views
Do all true number thesis with universal form allways has a proof？ [closed]
true number thesis with universal form: Goldbach conjecture, twin primes, every normal number thesis with a form ∀x∈N.P(x).
As a comment from André Nicolas, Matiyasevich's theorem(Hilbert's tenth ...
4
votes
1answer
68 views
How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?
I have two relations:
1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$.
2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$.
From these two how does it follow that ...
9
votes
0answers
131 views
A contest question
$p$ is an odd prime,denote $$f(x)=\sum_{k=0}^{p-1}\binom{2k}{k}^2x^k.$$
Prove that for every $x\in Z$,$$(-1)^\frac{p-1}2f(x)\equiv f(\frac{1}{16}-x)\pmod{p^2}.$$
This is a contest question,I do not ...
3
votes
1answer
67 views
Most elegant/simple proof of the irrationality of $\pi$
What is the most elegant/shortest proof of this? The proofs I have seen are quite long, unlike the proof of the irrationality of $e$.
thanks
4
votes
4answers
84 views
The elements of $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$
I'm really confused with this one...
How can I determine the elements of the module $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$? Or its cardinality?
Does ...
1
vote
0answers
38 views
How many “$1$” is need at least for a decimal number,which is consisting of “$0$” and “$1$” and divisible by $p$?
How many "$1$" is need at least for a decimal number,which is consisting of "$0$" and "$1$" and divisible by $p$?
If $p=2^k\cdot d+1$, and $10^d\equiv 1 \pmod p$,then $10^n\equiv -1 \pmod p$ has no ...
5
votes
2answers
77 views
When is it solvable:$10^a+10^b\equiv -1 \pmod p$
If $p$ is a prime, $(a,p)=1$,denote $ord(a,p)=d,$ where $d$ is the smallest positive integer solution to the equation $a^d\equiv 1 \pmod p$.We can prove that $$10^n\equiv -1 \pmod p\tag1$$ is ...
12
votes
2answers
96 views
Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$
How do I simplify:
$$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$
Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a ...
21
votes
2answers
321 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
69 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?
4
votes
1answer
62 views
Last non zero digit of $n!$ [duplicate]
What is the last non zero digit of $100!$?
Is there a method to do the same for $n!$?
All I know is that we can find the number of zeroes at the end using a certain formula.However I guess that's of ...
4
votes
2answers
62 views
For every prime of the form $2^{4n}+1$, 7 is a primitive root.
What I want to show is the following statement.
For every prime of the form $2^{4n}+1$, 7 is a primitive root.
What I get is that
$$7^{2^{k}}\equiv1\pmod{p}$$
...
2
votes
2answers
79 views
A binary quadratic form: $nx^2-y^2=2$
For which $n\in\mathbb{N}$ are there $(x, y)\in\mathbb{N}^2$ such that $nx^2-y^2=2$ ?
3
votes
2answers
72 views
Is there an algebraic number which has all possible combinations of numbers?
Today i saw this question. A similar question just came into my mind. Is there any irrational algebraic number so that it contains all possible number combinations in its digits? I'm really curious ...
1
vote
0answers
38 views
Extending a rational entry matrix to an orthogonal matrix.
Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the ...
0
votes
3answers
38 views
Square-Trangular Numbers Checking Answer
Problem: The first 2 numbers that are both squares and triangles are 1 and 36. Find the next one and if possible, the one after that.
Answer: 1225, 41616
Problem: Can you figure out an efficient way ...
2
votes
1answer
31 views
Question about the definition of representability of a quadratic form
Say I have an integral quadratic form $q$ on $\mathbb{Z}^r$ and another integral binary quadratic form $\tilde{q}$. What does it mean for $q$ to primitively represent $\tilde{q}$? I can't seem to find ...
13
votes
3answers
410 views
Fermat Last Theorem for non Integer Exponents
We now that Fermat's last theorem is true so there are not positive integer solutions to
$$x^n+y^n=z^n$$
for $n\in\mathbb{N}$ and $n>2$.
But what about if $n\in\mathbb{R}$ or $n\in\mathbb{R}^+$?
4
votes
1answer
62 views
$p^{3}+m^{2}$ is square of a number.
Well i thought it is a nice problem so i will post it here.
1) Prove that for every natural numbers $m$, There is at most two primes $p$ where $p^{3}+m^{2}$ is the square of a number.
2) Find all ...
1
vote
1answer
154 views
How to prove these two ways give the same numbers?
How to prove these two ways give the same numbers?
Way 1:
...
4
votes
0answers
48 views
Free algebra over $\mathbb{Z}/N\mathbb{Z}$
Let $A$ be a commutative finite free $\mathbb{Z}/N\mathbb{Z}$ algebra of rank 2 with unit discriminant.
I have two questions :
1) Why is it true that $A/pA$ is isomorphic to either ...
3
votes
1answer
59 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 ...
10
votes
2answers
200 views
$x^2+x+1$ is the cube of a prime.
Please help me find all natural numbers $x$ so that $x^2+x+1$ is the cube of a prime number.(Used in here)
3
votes
4answers
41 views
Proving $\frac{1}{2}(5x+4),\;2 < x,,\;\text{isPrime}(n)\Rightarrow n = 10k+7$
How is it possible to establish proof for the following statement?
$$n = \frac{1}{2}(5x+4),\;2<x,\;\text{isPrime}(n)\;\Rightarrow\;n=10k+7$$
Where $n,x,k$ are $\text{integers}$.
To be more ...
3
votes
1answer
44 views
Knowing that $n= 3598057$ is a product of two different prime numbers and that 20779 a square root of $1$ mod $n$, find prime factorization of $n$.
Knowing that $n= 3598057$ is a product of two different prime numbers and that 20779 is a square root of $1$ mod $n$, find prime factorization of $n$.
What I have done so far:
$n = p \cdot q$
...
4
votes
1answer
68 views
Property of natural numbers involving the sum of digits
How can you prove that every natural number $M$ or $M+1$ can be written as $k + \operatorname{Sum}(k)$, where $\operatorname{Sum}(k)$ represents the sum of the digits of some number k.
Example:
$$
...
0
votes
0answers
26 views
number property [duplicate]
How can you prove that every natural number M or M+1 can be written as k + Sum(k), where Sum(k) represents the digits sum of number k.
Example:
248 = 241 + Sum(241) = 241 + 2 + 4 + 1
-4
votes
0answers
97 views
Is not a perfect square
If $\frac{m+2}{2}<n<m$ where $m,n$ posivite integers , show that the $2^{2n-2}-2^m+1$ is not a perfect square.
The question changed!!!
4
votes
1answer
29 views
Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function
In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$.
I have a question about Theorem 2.27 on page 22.
My ...
