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
1answer
18 views
Integral basis for a number field
I need some help in solving the following problem:
Suppose $K$ is a number field and $K=\mathbb{Q}(\theta)$ where $\theta\in\mathfrak{O}_K$, the ring of integers of $K$. Now among the elements in ...
0
votes
0answers
13 views
The generating function for Bernoulli polynomials
The generating function for Bernoulli polynomials is given by:
$$\frac{ue^{ux}}{e^u-1}=\sum_{n\geq 0}B_n(x)\frac{u^n}{n!}$$
Now, I have the following expression:
...
1
vote
1answer
24 views
Identity involving complex sigma function
When trying around with the DivisorSigma function of Mathematica, I found this Identity:
$\#\{a\mid\exists b\in\mathbb{Z}[i]: ab=n\}=\underbrace{\#\{a\mid\exists ...
0
votes
0answers
42 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 ...
13
votes
3answers
555 views
Yitang Zhang: Prime Gaps
Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific.
EDIT$^1$:
Are there any experts here who can ...
1
vote
5answers
54 views
Show $60 \mid (a^4+59)$ if $\gcd(a,30)=1$
If $\gcd(a,30)=1$ then $60 \mid (a^4+59)$.
If $\gcd(a,30)=1$ then we would be trying to show $a^4\equiv 1 \mod{60}$ or $(a^2+1)(a+1)(a-1)\equiv 0 \mod{60}$. We know $a$ must be odd and so $(a+1)$ ...
2
votes
1answer
64 views
Sums of powers being powers of the sum
I'm looking for literature on solving problems of the form
$$
n_1^\alpha+\cdots+n_k^\alpha=(n_1+\cdots+n_k)^\beta
$$
for positive integers $n_1,\ldots,n_k$ and fixed parameters $k$ and ...
11
votes
0answers
73 views
How many points can you find on $y=x^2$, for $x \geq 0$, such that each pair of points has rational distance?
Open problem in Geometry/Number Theory. The real question here is:
Is there an infinite family of points on $y=x^2$, for $x \geq 0$, such that the distance between each pair is rational?
The ...
1
vote
2answers
51 views
Special binary string
Imagine a binary string of increasing length, up to infinity.
What makes it so special? Well, just a simple "rule":
for any given length (odd or even), if one folds the string in half, there is at ...
16
votes
1answer
159 views
Prove every integer exists in this sequence?
Please could someone give me a hint on this sequences question? The question is to prove that every integer appears infinitely many times in the following sequence:
$$ \pm 1^{2} , \pm 1^{2} \pm 2^{2} ...
5
votes
2answers
187 views
How to prove to be an irrational number? Like $\sqrt{2}$ $\sqrt{3}$ or $\sum\limits_{k=1}^{\infty} \frac{1}{n^2}=\pi^2/6$
As we know $\sqrt{2},\sqrt{3}$ are irrational numbers. And I see some proofs on the net.
So I doubt that how $e,\pi$ or already known irrational numbers are proved to be irrational.
In fact, I got ...
0
votes
1answer
26 views
Any way to simplify this gcd totient function
I have the following expression
$$\frac{gcd(a,b)}{\varphi(gcd(a,b))}$$
$a,b$ are known positive integers. Is there any way to rephrase this or simplify it?
7
votes
2answers
81 views
The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-(\frac{-ab}{p})$
What I need to show is that
For $\gcd(ab,p)=1$ and p is a prime,
the number of solutions of the equation $ax^2+by^2\equiv 1\pmod{p}$ is exactly $p-(\frac{-ab}{p})$.
I got a hint that I have to use ...
3
votes
1answer
86 views
Career in Number Theory?
I am about to get my B.S. in Mathematics, and I will be applying for PhD in pure mathematics next year, with future plans of teaching and doing research. Over the past year, I have developed a great ...
3
votes
4answers
114 views
Find $a,b,c \in \mathbb {Q}$
Find $a,b,c \in \mathbb {Q}$ such that:
$\left\{\begin{array}{rl} x^3&\in \mathbb Q \\ x&\notin \mathbb{Q}\\ ax^2+bx+c &=0\end{array}\right.$
I tried Vieta's formulas, but seem like it ...
0
votes
0answers
44 views
Amount of Background Needed for Number Theory Research
How much background is needed to do research pure number theory? I mean things like descriptions under 18.785 and 18.786 in http://student.mit.edu/catalog/m18b.html. I get the impression that it takes ...
4
votes
2answers
54 views
Number Theory $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd
If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.
Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with
...
3
votes
0answers
38 views
Multiplicatives [duplicate]
Let $f: N \to N$, $f(2) = 3$, and $f(ab) = f(a)f(b)$, that is, f is a multiplicative function. f is also strictly increasing. Show that no such function exists.
Progress: Apparently, this is proven ...
3
votes
1answer
43 views
System of Diophatine equations $x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ y \in \mathbb{Z^+}$
Let $a_1,\ a_2,\ a_3,\ \ldots,\ a_n$ be distinct positive integers.
Find $x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ y \in \mathbb{Z^+}$
such that: $$\left\{\begin{array}{rl}(x_1,x_2,\ldots,x_n)&=1\\ ...
2
votes
0answers
46 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
52 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
1answer
169 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
54 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
137 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
78 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
76 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
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 }={ ...
12
votes
3answers
305 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
65 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
83 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
42 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
38 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
44 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,
$$
...
-2
votes
1answer
126 views
Do all true number thesis with universal form allways has a proof？ [closed]
true number thesis with universal form: Goldbach conjecture, twin primes, every normal number thesis with a form ∀x∈N.P(x).
As a comment from André Nicolas, Matiyasevich's theorem(Hilbert's tenth ...
4
votes
1answer
73 views
How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?
I have two relations:
1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$.
2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$.
From these two how does it follow that ...
9
votes
0answers
145 views
A contest question
$p$ is an odd prime,denote $$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}.$$
This is a contest question,I do not ...
3
votes
1answer
68 views
Most elegant/simple proof of the irrationality of $\pi$
What is the most elegant/shortest proof of this? The proofs I have seen are quite long, unlike the proof of the irrationality of $e$.
thanks
4
votes
4answers
87 views
The elements of $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$
I'm really confused with this one...
How can I determine the elements of the module $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$? Or its cardinality?
Does ...
1
vote
0answers
38 views
How many “$1$” is need at least for a decimal number,which is consisting of “$0$” and “$1$” and divisible by $p$?
How many "$1$" is need at least for a decimal number,which is consisting of "$0$" and "$1$" and divisible by $p$?
If $p=2^k\cdot d+1$, and $10^d\equiv 1 \pmod p$,then $10^n\equiv -1 \pmod p$ has no ...
5
votes
2answers
77 views
When is it solvable:$10^a+10^b\equiv -1 \pmod p$
If $p$ is a prime, $(a,p)=1$,denote $ord(a,p)=d,$ where $d$ is the smallest positive integer solution to the equation $a^d\equiv 1 \pmod p$.We can prove that $$10^n\equiv -1 \pmod p\tag1$$ is ...
12
votes
2answers
99 views
Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$
How do I simplify:
$$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$
Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a ...
25
votes
2answers
425 views
Why is $\varphi$ called “the most irrational number”?
I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio ...
1
vote
1answer
69 views
Area of a circle is $A = \pi r^2$. Is it possible that both $A$ and $r$ are perfect integers.
Can you produce an example where both the area of a circle and it's radius are integers?
4
votes
1answer
63 views
Last non zero digit of $n!$ [duplicate]
What is the last non zero digit of $100!$?
Is there a method to do the same for $n!$?
All I know is that we can find the number of zeroes at the end using a certain formula.However I guess that's of ...
4
votes
2answers
64 views
For every prime of the form $2^{4n}+1$, 7 is a primitive root.
What I want to show is the following statement.
For every prime of the form $2^{4n}+1$, 7 is a primitive root.
What I get is that
$$7^{2^{k}}\equiv1\pmod{p}$$
...
2
votes
2answers
81 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
76 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
39 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 ...
