Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc.. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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Convergence of series involving Euler's totient function.

I have to show that if $\phi$ is Euler's totient function, then the series $\sum\limits_{n=2}^{\infty} \frac{1}{\phi (n) \log n}$ diverges and $\sum\limits_{n=2}^{\infty} \frac{1}{\phi(n) \log^2 n}$ ...
3
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0answers
19 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
5
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1answer
90 views

Prove by combinatorial method that $ \frac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $ is an integer

Prove that $$ \dfrac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $$ is a positive integer, where $(m,n) \in \mathbb{Z^{+}}$ I have already solved it using Legendre's Formula ...
2
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1answer
21 views

for any positive integer $a,b,n$,and $(a,b)=1$,Is $\frac{1}{a+b}+\frac{1}{a+2b}+\cdots+\frac{1}{a+nb}$ non integer,and How to prove that?

It's easy to prove that both $\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ and $\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2n+1}$ are nonintegers by multiply $2^k$and $3^k$, and how about the ...
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0answers
34 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
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15 views

On (known) applications of fixed point theorems to some conjectures in elementary number theory

Let $\sigma$ be the classical sum-of-divisors function. Call an integer $n$ almost perfect if $\sigma(n)=2n-1$. The only known examples are $n=2^k$ for $k \geq 0$. Let $I(n)=\sigma(n)/n$ be the ...
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1answer
54 views

Why are there so many conjectures in number theory and comparatively less in others?

My question is that : Why are there so many conjectures in elementary number theory and comparatively less in others? This is particularly weird because every topic in maths should have its equal ...
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1answer
56 views

Why is $\frac{4k -1}{2} \equiv 1 \pmod 4$?

Why is $\frac{4k -1}{2} \equiv 1 \pmod 4$? I need some help with understanding this... The original problem was: find the set of Numbers in which $ f(z)=\frac{1}{1- sin(z)} $ is not defined. This ...
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15 views

On Diophantine approximation and irrationality proofs

This question is an offshoot from this previous MSE post. I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / ...
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2answers
48 views

How to show that $(2, \sqrt{82})$ in $\mathbb{Z}[\sqrt{82}]$ is not pricipal?

I tried the obvious things, like using the norm and trying to show that there were no integer solutions to $a^2 - 82b^2 = 2$, but didn't get anywhere. (A friend asked me this.)
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2answers
64 views

Finding de Dr. Bronowski's number

This is a puzzle, created by Dr. J. Bronowski. All credit for this problem goes to him. Find the least positive integer such that moving the leading digit to the end produces a new integer that is ...
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1answer
26 views

unramified quadratic extensions of 2-adic numbers

i already know how to get the 7 quadratic extensions of $\mathbb{Q}_2$ from hensel's lemma. they are $\mathbb{Q}_2(\sqrt{d})$ for d = -10, -5, -2, -1, 2, 5, 10. question: which of these are ...
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1answer
64 views

Finding all primes $p$ such that $3p+20$ and $p+20 $ are primes

I need to find all primes $p$ such that $3p+20$ and $p+20 $ are primes. The first primes which satisfy the condition are: $3, 11, 17, 23$. I've tried to find the dependence, and found that $p \equiv ...
4
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0answers
47 views

theorem relating mersenne numbers?

For $(x2^9)^2=2^q-1+y^2q^2$,where $q$ is prime, is it possible to show that there exists only an unique solution for the pair $\{x,y\}$?
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1answer
37 views

Number of primes between $2k$ and $(\sqrt{k}-1)^2$.

I would like to prove the following. Let $\pi$ denote the prime counting function. Then for $k\geq 81$ we have $$ \pi(2k)-\pi\left(\left\lfloor(\sqrt{k}-1)^2\right\rfloor\right)\geq 6. $$ What I ...
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2answers
81 views

$3^x + 4^y = 5^z$ [duplicate]

This is an advanced high-school problem. Find all natural $x,y$, and $z$ such that $3^x + 4^y = 5^z$. The only obvious solution I can see is $x=y=z=2$. Are there any other solutions?
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0answers
22 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general?

(Note: This question has been cross-posted from MO.) Let $\sigma$ be the classical sum-of-divisors function. A number is said to be perfect if $\sigma(N)=2N$. If $q^k n^2$ is an odd perfect number ...
5
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1answer
49 views

A number-theoretic random walk on the integers

Suppose a random walker starts at $S_0 = 2$, and walks according to the following transition probabilities: If the walker is on the $n$th prime number $p_n$, she moves to either $p_n + 1$ or ...
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0answers
18 views

Is there a classification of perfect squares in the rings of cyclotomic integers?

We know that perfect squares in $\mathbb {Z} $ has the form $n^2, n\in \mathbb {Z} $. More generally, can I use a nice condition to find a solution of Diophantine equations of the form $x^2=c $, $c ...
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0answers
43 views

Proving the existence of a sequence such that

I am trying to prove that there exists a sequence, for example: $$ f(n) = n! $$ (or we can select any sequence we need to prove the existence of just one), with the following property: edit: for ...
2
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2answers
49 views

Proof that $a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$

I need to prove that: $$a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$$ What I thought was: $$a\mid x \implies x = aq_1\\b\mid x\implies x = bq_1$$ Also, since $\gcd(a,b) = 1$, we have that ...
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2answers
62 views

$\lfloor 2x \rfloor \lfloor 3x \rfloor$ [on hold]

From $1$ to $10000$ including both, how many of those integers can be written as: $$\lfloor 2x \rfloor \lfloor 3x \rfloor$$ Where $x$ is a real number?
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totient function and inclusion-exclusion principle

How can one prove the property established by Gauss $$ \sum_{d\mid n} \varphi(d)=n$$ using the inclusion–exclusion principle? I was thinking to use that with the same method one can prove that ...
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3answers
71 views

Show $ x^2 = 1 + y^2 + z^2$ has infinitely many solutions [on hold]

Show $ x^2 = 1 + y^2 + z^2$ has infinitely many solutions Can anyone give me the specific steps for this problem?
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2answers
78 views

Finding two solutions to $x^2 - 6y^2 = 1$ using continued fractions [on hold]

Can anyone show me how to find the solutions to $x^2-6y^2=1$ by using continued fractions? I know how to find the fractions for $\sqrt6$ but do not know how to proceed. THANK YOU!!!
5
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3answers
920 views

Where is the mistake in this proof about rationals having eventually periodic decimal expansions?

I know that a rational number has an eventually periodic decimal expansion, and not necessarily just periodic. So what is wrong with this 'proof' that any rational number has a periodic decimal ...
0
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0answers
90 views

ABC triples when C is a power. [on hold]

For any integer $x$ we can apparently always find a sufficiently large $y$ such that there is an $ABC$ triple $A + B = C$ (from the $ABC$ conjecture $rad(ABC) < C$) where $C = x^y$. The minimum ...
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2answers
49 views

A question about Quadratic residue

I need help with this question : Prove that for each prime number p there exist $a,b \in Z$ such that $-1\equiv a^{2}+b^{2}\pmod p $ When $p\equiv1\pmod4$ it is easy because -1 is a quadratic ...
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2answers
52 views

How many pairs of adjacent bits? [on hold]

Every decimal number has a binary representation which is actually a string of bits. If a bit of a number is $1$ and the next bit is also $1$, then we can say that the number has $1$ pair of adjacent ...
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1answer
57 views

Find $f(2015)$ in function $f$ defined below

Let $\mathbb{S}$ be the set $\mathbb{R}^+ \cup \{0\}$ Let a function $f:\mathbb{S} \rightarrow \mathbb{S} $ be defined as: $$f(x^2+y^2) = y^2f(x)+x^2f(y) +x^4+y^4$$ If done so, then what would be ...
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1answer
23 views

Proving $\gcd(N^a-1,N^b-1)=N^{\gcd(a,b)}-1$.

I have come by one solution only, but things were derived too quickly without me understanding how or why. How does knowing that $\gcd(a,b)$ is a factor and a and b, actually derive that ...
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1answer
26 views

Show that $a^{p+1\over 4}$ solves the equation $x^2 ≡ a \pmod p$.

Let $p \equiv3 \pmod 4$ be a prime number, and let $1 \le a\le p − 1$ be a quadratic residue. Show that $a^{p+1\over 4}$ solves the equation $x^2 ≡ a \pmod p$. I know that if $(a,n)=1$ and $p\ge ...
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0answers
30 views

Roadmap to $p$-adic numbers: where a self-learner should look for references

TL;DR at the end of the question. I’m currently trying to learn as much as possible about p-adic numbers. I’m not sure what is the most fascinating part of the theory, but the use of the adjective ...
1
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1answer
35 views

RSA fixed point

What is the number of RSA fixed points, in other words how many $m$ are there such that $$m^e\equiv m \pmod{n}$$ where $n=pq$, for primes $p,q$. I know that the answer is ...
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0answers
58 views
+50

Does there exists a positive $t$ that satisfy this given condition?

I am curious about the validity of my claim concerning the equations: $(2k-1)t+1$ (1) $(2k^2-2k)t+(2k-1)$ (2) where $k=2,3,4,...$ My claim is for almost all $k$ or for infinitely many $k$, there ...
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0answers
30 views

a question in De koninck and luca's analytic number theory [on hold]

what is your idea,can you introduce a book for these kind of problems?
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4answers
65 views

Product of two integers of the form $x^2+my^2$ is of the same form.

Let $x,y,a,b\in \mathbb Z$. Prove that there are integers $c$ and $d$ so that \begin{equation*} (x^2+y^2m)(a^2+b^2m)=c^2+d^2m. \end{equation*} I'm stuck, I took the product and got ...
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1answer
15 views

List products in descending order

I'm doing a problem from Project Euler where I need to examine a descending order of multiples of 3 digit numbers ...
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0answers
18 views

Is there a match between this modified prime pi function and the Log integral function?

Table T is defined as through the properties that accumulated row sums give prime numbers, while accumulated column sums give composite numbers. ...
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2answers
67 views

Elliptic curve $y^2= x^3 + x$ over the finite field $\mathbb{F}_p$ with $p \geq 3$.

Consider the elliptic curve $$E: y^2= x^3 + x$$ over the finite field $\mathbb{F}_p$ with $p \geq 3$. I want to show that $|E(\mathbb{F}_p)| \equiv 0 \mod 4$. I know that, if $p \equiv 3\mod 4$, ...
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4answers
61 views

Proving that $\varphi(n)=n\prod (1-1/p)$ without using multiplicativity

$$\varphi(n)=n\prod_{p \ \text{prime}} (1-1/p)$$ Can this useful formula be derived without using the fact that Euler's totient function is multiplicative?
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42 views

Is the Ulam Spiral just a coincidence?

I was messing around with the Ulam spiral because I was a little skeptical on it having any actual relevance. I noticed that if you lay out the spiral and then circle all the even numbers, it displays ...
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0answers
63 views

Simple $\{-1,0,1\}$ equation set

I'm trying to find the shortest path, getting from $x=0$ to $x=k$ in a certain problem, where I can slowly accelerate and decelerate. It comes down to finding the smallest $n$ and set of values ...
3
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0answers
35 views

Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
3
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1answer
73 views

The elliptic curve $y^2 = x^3 + 2015x - 2015$ over $\mathbb{Q}$

Consider the elliptic curve \begin{equation*} E: y^2 = x^3 + 2015x - 2015~\text{over}~\mathbb{Q}. \end{equation*} I want to prove that $|E(\mathbb{F}_7)| = 12$, that $|E(\mathbb{F}_{19})| = 19$ and ...
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1answer
19 views

Solve quadratic congruences.

$x^2= -1 (mod 43)$. I want know How to solve this problem and also what will be the general approach for solving any of such quadratic congruences.
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2answers
57 views

Finding the solutions of $x^2\equiv 9 \pmod {256}$.

Find the solutions of $x^2\equiv 9 \pmod {256}$. I try to follow an algorithm shown us in class, but I am having troubles doing so. First I have to check how many solutions there are. Since $9\equiv 1 ...
2
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2answers
90 views

Prove an inequality of Prime Numbers

The problem on which I am currently stuck is, Is it true that, $$x+y< \dfrac{p_{\pi(x)}+p_{\pi(y)}+p_{\pi(x)+1}+p_{\pi(y)+1}-2}{2}$$ for all sufficiently large $x$ and $y$, $x+y$ is a prime ...
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0answers
23 views

How to find a set of integer vectors (of length L) such that all its subsets with size L are linearly independent?

Given a number $M\geq L$, how to find a set of $M$ vectors in $\mathbb{Z}_{\geq0}^{L}$, say $S=\{\mathbf{a}_1,\cdots,\mathbf{a}_L\}$, such that: 1-All subsets of $S$ with size $L$ are linearly ...
0
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1answer
20 views

Is $(a^s\pmod{n})^k = a^{sk}\pmod{n}$?

Is $(a^s\pmod{n})^k = a^{sk}\pmod{n}$? And if it is, how come? I've thought about it and this is the only thing that makes sense (for now).