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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Prove that there are infinitely many composite numbers so that $\phi (n)$ divides $n-1$

Prove that there are infinitely many composite numbers so that $\phi (n)$ divides $n-1$, where $\phi (n)$ is Euler's function. I really don't know how to solve this, so could someone give me a hint or ...
0
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1answer
55 views

Solution of $8x^2+x+1=0\in \mathbb{Q}_2$.

We are asked to determine whether the equation $f(x)=8x^2+ x+ 1$ has a root in $\mathbb{Q}_2$. Now, I immediately think to apply Hensel's Lemma, with $\alpha=1$, by which I mean $f(\alpha)\equiv 0 ...
7
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1answer
71 views

On prime factors with $n^2+n+1$

Show that: There are infinitely many positive integers $n$ such that all prime divisors of $n^2+n+1$ are not greater than $c\cdot n^{0.8}$, where $c$ is constant. Maybe this $0.8$ is not best ...
14
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3answers
780 views

Transcendental number

While reading on Wikipedia about transcendental numbers, i asked myself: Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e, \frac{\pi}{e}$ etc. are transcendental numbers? ...
3
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1answer
58 views

Is it known whether any positive integer can be written as the sum of $n$ different squares?

Is it known whether any sufficiently large positive integer can be written as the sum of four different squares? I know that every positive integer can be written as the sum of four not necessarily ...
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1answer
25 views

Proving that $n|x^ {φ(n)/2} − 1$ for every $x$ coprime to $n$.

Let $n \in \Bbb{N}$ for which there exist two coprime numbers bigger than 2 dividing n. Show that for every x coprime to n we have $n|x^ {\phi(n)/2} − 1$. Conclude that there is no primitive root ...
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0answers
13 views

Are integral combination and linear combination the one and the same in the field of number theory?

My question is a trivial question as to the exactness of the meaning of integral and linear in the study of the number theory. Do they hold the same meaning?
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2answers
11 views

Proof of show transitivity between 3 variables with exponents

If $a^5$ divides $b$ and $b^5$ divides $c,$ show that $a^{20}$ divides $c.$ Please help me prove this proposition.
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0answers
18 views

Lists of negative discriminants by class group?

Is there a handy listing of the discriminants of imaginary quadratic fields having a given ideal class group? It would be nice to use such a resource as a source of examples. For example, we're all ...
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0answers
14 views

What are the elements with the minimal absolute values in cyclotomic extensions of the integer ring?

We can easily see 1 has the (nonzero) minimal absolute value in the integer ring. In the Gaussian integer ring and the Eisenstein integer ring, 1 is also the minimal (complex) absolute value. However, ...
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0answers
23 views

Approximating the inverse logarithmic integral

Numerical evidence suggests that \begin{align} &\operatorname{li}^{-1}(n)=x_n+\mathcal{O}\left(\log^2 n\right)\\ \end{align} where \begin{align} &x_n=x_{n-1}+\log \left(x_{n-1}\right)\\ ...
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0answers
17 views

Is there a multi-perfect number which is NOT of the form $2^n(2^{n+1}-1)u$?

The even perfect numbers have the form $2^n(2^{n+1}-1)$, where $2^{n+1}-1$ is a (Mersenne-)prime. Is every multi-perfect number of the form $2^n(2^{n+1}-1)u$, where $u$ is some odd number ? It is ...
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0answers
21 views

Is it conjectured that there are no odd multi-perfect numbers?

It is conjectured that there is no odd perfect number. But is there a stronger conjecture that there are no odd multi-perfect numbers ? Wikipedia shows a useful link, but my conjecture is not ...
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2answers
45 views

2 questions in Number Theory about primitive roots/quadratic residue

I tried to solve this 2 questions but without a success: Is $13$ a sixth power modulo $289$? Find all the solutions of $x^{8}\equiv 3\mod 13$ In question 1, I tried to see if $13$ is a quadratic ...
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1answer
22 views

Continued Fractions, Euclid's Algorithm

I know how to express $45/17$ as a continued fraction using Euclid's algorithm. But how do i go about expressing $17/45$ as a continued fraction? I think I worked it out, is it [0,2,1,1,1,5]?
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2answers
29 views

Conductor of a ring

An easy (possibly trivial) question from Neukirch's Algebraic Number Theory, p.47 Let $A$ be a Dedekind domain, $K$ its fraction field, $L$ a finite separable extension of $K$ and $B$ the integral ...
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1answer
21 views

Given Definitions Prove about Square-Free numbers, etc.

Definition 1 For any number $x$, $N_j(x)$ is the number of positive integers less than or equal to $x$ that have all their prime divisors among the set of the first $j$ primes ...
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1answer
38 views

examine and prove the following statements including $\varphi$

How do I show that $\varphi (m)= \sum \limits_{k=1}^m \lfloor 1/(k,m)\rfloor $. How do I show that: $\sum \limits_{k=1}^m \varphi(k) \lfloor m/k \rfloor = m(m+1)/2$ I did not have any particular ...
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3answers
40 views

Divisibility number theory problem

How many $k,m$ exist such that $ \frac {k^2+m^2}{2(k-m)}$ is also an integer. $k,m \in \mathbb {Z} ^ + $ My guess that there is finitely many solutions but I can't seem to be able to prove so.
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2answers
423 views

Application of Taylor's Theorem in Number Theory

I'm working through Alan Baker's book A Concise Introduction to the Theory of Numbers, and there's an assertion in there that confuses me. Here's the quote: It is easily seen that no polynomial ...
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2answers
38 views

An infinite square arithmetic progression? [duplicate]

How to prove that there does not exist and infinite arithmetic sequence that all of it's terms are distinct squares of integers?
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2answers
35 views

A extending the p-adic valuation to a quadratic extension of $\mathbb{Q}_p$

I'm trying to solve the following problem. Prove that, if $d \in \mathbb{Z}_p$ is non-square, then $|a + b \sqrt{d}|p = |a^2 − b^2d|^{1/2}_p$ , for any $a, b \in \mathbb{Q}p$, defines a ...
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0answers
22 views

Is it possible to make uniform irrational numbers from nonuniform irrationals? [closed]

If you were given a nonuniform irrational number is it possible to shuffle it with addition and multiplication of integers? Edit: A nonuniform irrational number is an irrational number whose digits ...
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3answers
91 views

Confusion Number theory question

Prove for each positive integer $n$, there exists $n$ consecutive positive integers none of which is an integral power of a prime number. I'm not getting a single idea of how to approach it. One I ...
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2answers
32 views

A question on perfect square

Prove that if $ab$ is a perfect square and $\gcd(a,b)=1$, then both $a$ and $b$ must be perfect squares. Their Answer: Consider the prime factorization $ab=p_1^{e_1}\cdots p_k^{e_k}$. If $ab$ ...
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2answers
26 views

Topology of $\Bbb{Q}_p$

Let $a\in \Bbb{Q}_p$. Is $ a+p^x\Bbb{Z}_p$ an open set around $a$ in the topology of $\Bbb{Q}_p$. Here $x \in \Bbb{Z}$. Also I have another question. Is $\mathbb{Z}_p$ open in $\Bbb{Q}_p$?
3
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1answer
56 views

Find all pairs of primes $p,q$ such that $pq \mid 2^p +2^q$

Find all pairs of primes $p,q$ such that $pq \mid 2^p +2^q$. My attempt : When either one of them is $2$ then easy case checking gives me set of solutions. But what happens when neither of them is ...
7
votes
2answers
159 views

Integer solutions of $x^3 = 7y^3 + 6 y^2+2 y$?

Does the equation $$x^3 = 7y^3 + 6 y^2+2 y\tag{1}$$ have any positive integer solutions? This is equivalent to a conjecture about OEIS sequence A245624. Maple tells me this is a curve of genus $1$, ...
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0answers
8 views

number of distinct numbers of the form $e(k^2(4a)^{-1})$

Let $q$ be a large prime. Define $e(n)=\exp\{2\pi i\frac{n}{q}\}$. What is the cardinality of the set $\{e(k^2(4a)^{-1}): a,k\in\mathbb{N}\}$? Here $a^{-1}$ means the multiplicative inverse of $a$ in ...
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3answers
52 views

Why Sum $> 1/2$ in proving reciprocal of prime diverges

In my number theory book it says that to show that the sum of the reciprocals of the primes diverges, it’s enough to show that, for any $j$: ...
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0answers
26 views

Number Theory proof with parts : $N_j(x)$ is the number of positive integers less than or equal to $x$ [closed]

Definition 1. For any number $x$, $N_j(x)$ is the number of positive integers less than or equal to $x$ that have all their prime divisors among the set of the first $j$ primes ...
0
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2answers
30 views

(Help) Find integers x <17 and y < 17 that satisfy: 2x + y ≡ 4 (mod17) 5x−5y ≡ 9 (mod17) [closed]

I don't know where to start, can anyone help with how to start this question: Find integers $x <17$ and $y < 17$ that satisfy: $$\eqalign{ 2x + y &\equiv 4 \pmod{17}\cr 5x−5y ...
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0answers
51 views

Singularities in the weighted projective space

Is there an explicit criterion for checking that a hypersurface $f=0$ of degree $d$ and in $\mathbb{P}(a_0,\ldots,a_n)$ is smooth ? I could not convince myself that the criterion $\nabla f\neq 0$ ...
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3answers
44 views

$N^{1/2}$ and randomness

I apologize if this question is overly vague, but part of the reason I am asking is because I don't know a more precise way of discussing these ideas. To state a general question: What, if any, ...
3
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1answer
27 views

Asymptotic for primitive sums of two squares

A positive integer $n$ can be written primitively as the sum of two squares, meaning $n = x^2 + y^2$ with $\gcd(x,y)=1,$ precisely when $n$ is not divisible by $4$ or by any prime $q \equiv 3 \pmod ...
1
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1answer
22 views

Least sum of products of powers

Numbers from the set $\{2^1, 2^2, ..., 2^{10}\}$ are somehow permuted and paired with numbers from the set $\{3^1, 3^2, ..., 3^{10}\}$. Numbers in each pair are multiplied and the products are summed. ...
0
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1answer
32 views

Division of $t^a-1$ by $t^b-1$

If $a,b\in \mathbb{N}$ with $b\neq 0$ and $r$ is the remainder of $a$ when divided by $b$, how do you show that for all integers $t>1$ the remainder of $t^a-1$ when divided by $t^b-1$ is $t^r-1$? ...
0
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1answer
13 views

What are the benefits of using reduction map and lift instead of function and inverse image?

I'm reading William Stein's: Elementary Number Theory: Primes, Congruences, and Secrets. And I found this definition. It employs the concept of reduction map and lift, but it seems to be very ...
2
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1answer
26 views

Incorrect Euler Totient Function definition?

According to wikipedia, definition of Euler's totient function (or Euler's totient function) is: Euler's totient function is an arithmetic function that counts the positive integers less than or ...
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0answers
74 views

How to show that the equation $7^3d^2-3^3c^2=1$ has infinitely many integer solutions ?

How to show that the equation $7^3d^2-3^3c^2=1$ has infinitely many integer solutions ? Please help . Thanks in advance
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2answers
56 views

Does the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ?

How to show that the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ? ( If this can be shown then solutions of $12x^2-8y^2=4$ give infinitely many powerful numbers ...
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1answer
30 views

Ramification in $\mathbb Q(\zeta_5, \sqrt[5]2)/\mathbb Q(\zeta_5)$

Let $F=\mathbb Q(\zeta_5,\sqrt[5]2)$ and $K=\mathbb Q$ where $\zeta_5$ is a primitive $5$th root of unity and let $p=73$ be a prime in $K$. Fix primes $\mathfrak p$ and $\mathfrak q$ above $73$ in ...
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2answers
29 views

quadratic reciprocity

I know $x^2\equiv-7\pmod7$ has solutions. How can I check if $x^2\equiv-7\pmod{49}$ has solutions? I know $-7\equiv42\pmod{49}$ but $49$ isn't a prime so I can't use Euler's criterion. How shall I do ...
3
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1answer
64 views

The number of primes in an interval

What is the smallest known $c$ so that for any $n\geq 2$ there are at least $n/\log_2{n}$ primes between $n$ and $cn$ (inclusive)? The prime number theorem seems to give an asymptotic result so I am ...
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0answers
24 views

Could you please explain the algorithm for the below given number series generated by Excel Fill series?

Below images represent the number series that are obtained using the Excel Fill Series If you input 13, 16, 17 then 19.33333, 21.33333, 23.33333, 25.33333,.. is generated. If you input 34, 424, ...
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0answers
118 views

Whether a functor is exact?

I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where ...
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2answers
24 views

Have I solved this congruence correctly?

![] [1]: http://i.stack.imgur.com/UMDnZ.jpg [1] Which can not be solved as gcd(16,22)=2, 2 does not divide 7. Hence no solutions to the congruence. Furthermore the congruence can not be solved as ...
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2answers
30 views

Equality symbols in modular arithmetic

E.g., can I write $(a^{p})^{2p} \equiv a^{2p}=a^pa^p\equiv aa\equiv a^2\pmod{\! p}$? I often see equality symbols inbetween mod equivalences. The equality signs point out the equality is not ...
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0answers
16 views

The lower bound of the number of cubic residue mod n. [duplicate]

For arbitrary positive integer $n$ , Denote $a\sim_n b \iff a^3\equiv b^3 \mod n$, and $P(n):=\mathrm{Card}\{\mathbb{Z}/\sim_n\}$, How to calculate the value ...
0
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1answer
26 views

$\mathbb{Z}$-basis of quadratic ring

Definition. A quadratic ring $R$ is a commutative ring with $(R,+) \cong \mathbb{Z}^2$ (The additive abelian group of $R$ is isomorphic to $\mathbb{Z}^2)$ Lemma. If $R$ is a quadratic ring, ...