Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.
1
vote
1answer
30 views
$2^{q-1}\equiv 1\pmod{q}.$
The question is asking to show that $q$ must be prime given
$$ 2^{q-1}\equiv 1\pmod{q}. $$
1
vote
2answers
30 views
Inverse | Modulo | Power
Describe the inverse of $5$ modulo $18$ as a positive power of $5\pmod{18}$.
I've got that the inverse of $5$ is $11$, but is this question asking to find a $t$ such that
$$ 11=5^t\pmod{18}?$$
2
votes
0answers
32 views
Distribution of Digit Products
A digit product $P(n)$ of a natural number $n$ is given by the product of its decimal digits. For example:
$$P(1234) = 24,\;\;\; P(24) = 8,\;\;\; P(8) = 8$$
$$1\times2\times3\times4 = 24, \;\;\; ...
1
vote
0answers
24 views
Showing an elliptic curve has infinitely many points over $\mathbb{Z}_p$
I stumbled upon this question, and I can't think of how to do it, or what kind of results to use. The question is as follows:
Let $$y^2=x^3+ax+b$$ be an elliptic curve ($a,b$ integers), and let $p ...
1
vote
1answer
37 views
REVISTED$^1$ - Order: Modular Arithmetic
Relevant Literature:
Question:
Observe that $2^{10}=1024≡−1 \pmod{25}$.Find the order of $2$ modulo $25$.
Thoughts:
Direct answers are OK, but I'd like to know if I'm right that what I'm really ...
5
votes
1answer
59 views
Proof of $\lim_{s\to 1^+}\frac{\sum_p p^{-s}}{\ln(s-1)}=-1$
I am looking for a proof of $$\lim_{s\to 1^+}\frac{\sum_p p^{-s}}{\ln(s-1)}=-1.$$
If the proof is too long, a direct reference is fine. Here the sum $\sum_p$ denotes the sum over all prime numbers.
...
2
votes
2answers
50 views
Can't understand homework assignment
Let $p\in\mathbb{N}$ be an odd prime. Prove that if $p = 3\ ( mod\ 4)$ then $−1$ is not a square modulo $p$.
$\textbf{Hint}$ : recall that $\mathbb{Z}/p\mathbb{Z}$ is a field, so that its ...
0
votes
0answers
24 views
Which numbers of [0,1) have a unique base g expansion?
Good evening,
i know that is question is rather standard, but unfornunately I have not much knowledge of number theory.
Take $2 \leq g\in \mathbb{N}$. I know that every $x \in [0,1)$ can be ...
0
votes
3answers
96 views
A proof of $n*0=0$?
The only proof I've seen for this assumes that $0$ follows all the rules of arithmetic. How can we make that assumption when dividing by $0$ is a problem? I know that some people don't agree that all ...
2
votes
0answers
52 views
Can an odd perfect number be divisible by $165$?
I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $165$ (this is actually a stronger statement).
1
vote
1answer
21 views
Prove that if $n\geq\text{lcm}(a,b)$ and $\gcd(a,b)|n$ then $n=xa+yb$ for some integers $x,y\geq 0$
I thought I had it, but then I realized I didn't. Even just a hint—am I going in the right direction or should I try something completely different?
We know that $\gcd(a,b)=wa+zb$ for some integers ...
0
votes
1answer
35 views
Solving complex linear congruences
Find $x \in \mathbb{Z}[i]$ such that:
$(1+2i)x \equiv 1 \mod 3+3i$
How would you go about doing this? Best I can think of is keep guessing....
4
votes
5answers
47 views
Proving the remainder is $1$ if the square of a prime is divided by $12$
Given, $p$ is a prime number and $p>3$. How do we prove that the remainder $r$ is always $1$ if $p^2$ is divided by $12$?
2
votes
0answers
37 views
Family of elliptic curves with trivial torsion
I'm wondering, if it is true that the torsion subgroup of $y^2=x^3+p$ (for $p$ some prime, greater than 2), is always trivial?.
I was trying to prove this using Lutz-Nagell, but I can't quite get it.
...
2
votes
1answer
35 views
Lattices in $\mathbb{C}$
I have the following assignment:
consider the map
$$|\cdot|:\mathbb{Z}[i]\longrightarrow \mathbb{N},\qquad |a+ib|:=a^2+b^2$$
1) Prove that $|\alpha|<|\beta|$ iff $|\alpha|\leq |\beta|-1$ and ...
1
vote
4answers
44 views
Bertrand's postulate in another point of view
I was just wondering why can't we use prime number's theorem to prove Bertrand's postulate.We know that if we show that for all natural numbers $n>2, \pi(2n)-\pi(n)>0$ we are done. Why can't it ...
2
votes
1answer
49 views
Infinite prime numbers from a sum of powers
I am not sure if it's possible to get infinite prime numbers from this sum:
$$p=k^j+j^k$$
with $j\in\mathbb{N}, k\in\mathbb{N}$
I tried for $j=1,2,...9,k=1,2,...9$ and I get only eleven prime numbers.
...
2
votes
0answers
23 views
decomposition according to embeddings
Let $V$ a finite dimensional vector space over $\mathbb{Q}$ and let $F$ be a number field. Assume that there is an injective morphism of rings $F \hookrightarrow End(V)$. I would like to understand ...
5
votes
2answers
84 views
Is there a procedure to solve Diophantine Equations?
How would you go about solving a multivariable, non-linear Diophantine Equation?
1
vote
1answer
34 views
Frobenius element in cyclotomic extension
Let $K=\mathbb{Q}(\zeta_m)$. Then if $p\nmid m$ is any odd prime, how i can show that Frobenius map is
$(p,K/\mathbb{Q})(\zeta_m)=\zeta_m^p$.
We know, if $P$ is a prime above $p$
...
0
votes
0answers
17 views
Computability of division of large numbers
What is the largest computable mathematical division in terms of the number of digits that can be handled by a typical desktop computer using the best available big number libraries, assuming input is ...
5
votes
1answer
37 views
Image of the Norm on a Finite Dimensional Extension of $\mathbb{Q}_p$
I've been trying to see whether following assertion is true in order to give a quick proof of another problem I was doing: if $K$ is a finite dimensional extension of the $p$-adic numbers ...
1
vote
1answer
43 views
Proving that Bombieri's Theorem implies Linnik's theorem
I'm stuck on a line in the proof of Bombieri implies Linnik, where
Bombieri:
For primitive $\chi$ mod $q$ with $q \leq T$ we define $$N(\alpha, T; \chi)=\#\{\rho=\beta+i\gamma \;:\; ...
4
votes
1answer
49 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
26 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
28 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
48 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 ...
18
votes
3answers
2k 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
60 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 $(a+1)$ ...
3
votes
2answers
98 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 ...
13
votes
0answers
116 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
52 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
160 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:
$$ \pm 1^{2} , \pm 1^{2} \pm 2^{2} ...
5
votes
2answers
194 views
How to prove to be an irrational number? Like $\sqrt{2}$ $\sqrt{3}$ or $\sum\limits_{k=1}^{\infty} \frac{1}{n^2}=\pi^2/6$
As we know $\sqrt{2},\sqrt{3}$ are irrational numbers. And I see some proofs on the net.
So I doubt that how $e,\pi$ or already known irrational numbers are proved to be irrational.
In fact, I got ...
0
votes
1answer
27 views
Any way to simplify this gcd totient function
I have the following expression
$$\frac{gcd(a,b)}{\varphi(gcd(a,b))}$$
$a,b$ are known positive integers. Is there any way to rephrase this or simplify it?
7
votes
2answers
83 views
The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-(\frac{-ab}{p})$
What I need to show is that
For $\gcd(ab,p)=1$ and p is a prime,
the number of solutions of the equation $ax^2+by^2\equiv 1\pmod{p}$ is exactly $p-(\frac{-ab}{p})$.
I got a hint that I have to use ...
3
votes
1answer
91 views
Career in Number Theory?
I am about to get my B.S. in Mathematics, and I will be applying for PhD in pure mathematics next year, with future plans of teaching and doing research. Over the past year, I have developed a great ...
3
votes
4answers
116 views
Find $a,b,c \in \mathbb {Q}$
Find $a,b,c \in \mathbb {Q}$ such that:
$\left\{\begin{array}{rl} x^3&\in \mathbb Q \\ x&\notin \mathbb{Q}\\ ax^2+bx+c &=0\end{array}\right.$
I tried Vieta's formulas, but seem like it ...
0
votes
0answers
45 views
Amount of Background Needed for Number Theory Research
How much background is needed to do research pure number theory? I mean things like descriptions under 18.785 and 18.786 in http://student.mit.edu/catalog/m18b.html. I get the impression that it takes ...
4
votes
2answers
56 views
Number Theory $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd
If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.
Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with
...
3
votes
0answers
38 views
Multiplicatives [duplicate]
Let $f: N \to N$, $f(2) = 3$, and $f(ab) = f(a)f(b)$, that is, f is a multiplicative function. f is also strictly increasing. Show that no such function exists.
Progress: Apparently, this is proven ...
3
votes
1answer
43 views
System of Diophatine equations $x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ y \in \mathbb{Z^+}$
Let $a_1,\ a_2,\ a_3,\ \ldots,\ a_n$ be distinct positive integers.
Find $x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ y \in \mathbb{Z^+}$
such that: $$\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
53 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 ...
12
votes
1answer
196 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
49 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
142 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
77 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 ...
