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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2
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23 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 ...
1
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2answers
46 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 ...
2
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1answer
24 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]?
1
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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 ...
1
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1answer
22 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 ...
1
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1answer
42 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 ...
1
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3answers
49 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.
10
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2answers
430 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 ...
0
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2answers
40 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?
2
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2answers
36 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 ...
1
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3answers
93 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
33 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
30 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
votes
1answer
57 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
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2answers
164 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 $1$, ...
0
votes
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 $j$: ...
0
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0answers
73 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 $\nabla f\neq 0$ ...
1
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3answers
45 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
votes
1answer
29 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
23 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
votes
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 ...
0
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0answers
75 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 ...
1
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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 ...
1
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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
votes
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 ...
0
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0answers
27 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, ...
8
votes
0answers
123 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 ...
0
votes
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
32 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, ...
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1answer
34 views

Recursive Hexagon Problem to find number of hexagons at each stage

The source of this problem is this SPOJ question. Let me simplify it: A valid beehive is recursively defined as follows: 1. A single regular hexagon is a valid beehive. 2. To all the external cells of ...
2
votes
1answer
26 views

Can a set of integers be linearly indepedent over rational field $\mathbb{Q}$?

As title says, can a set of integers be linearly independent over rational field $\mathbb{Q}$ or integer ring $\mathbb{Z}$?
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2answers
23 views

Minimal way to choose set $X = \{x_1,..x_n\}$ such that $\sum_{i=1}^n x_i$ is not other sums of $n$ numbers in $X$

Let there be a set $X = \{x_1,\cdots, x_n\}$. I want $\sum_{i=1}^{n} x_i$ to be a unique sum in the sense that it cannot be represented by other sums of $n$ numbers in $X$ that involve at least one ...
3
votes
1answer
47 views

Cubic Congruence Solutions

While I was reading a paper on number theory, there was a claim which wasn't prove there and I couldn't find a way to justify it. The claim is as follows For a prime $p$, when $p\nmid a$, the number ...
1
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2answers
31 views

Every superperfect number(except $2$) is a square number?

I recently read about superperfect numbers: $σ^2(n) = 2n$, where $σ(n)$ is the divisor function. I saw that the first few numbers were: $2, 4, 16, 64, 4096, 65536, 262144$, which are all ...
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0answers
29 views

Ideal classes of $\mathcal{O}_k$

How can I show if two ideals are in the same ideal class when considering the ideal classes of $\mathcal{O}_k$? Could someone show me an example of discarding ideals that have been used before? Thank ...
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1answer
29 views

Continued Fraction Expansions Confusion

Let $\theta$ be an irrational number with continued fraction expansion $[a_0; a_1, a_2, \cdots]$. Suppose $P_n/Q_n = [a_0; a_1, \cdots , a_n]$ is the $n^{th}$ convergent. Then how do I show that ...
1
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1answer
21 views

Help finishing proof with polynomial discriminant?

Prove that the discriminant of $$f(x) = x^n + nx^{n-1} + n(n-1)x^{n-2} + \cdots + n(n-1)\ldots (3)(2)x + n!$$ is $(-1)^{n(n-1)/2}(n!)^n$. So far, I let $\alpha_1,\ldots, \alpha_n$ be the roots of ...
6
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4answers
769 views

Correct statement of Fermat's Last Theorem

I'm looking at the wikipedia page on Fermat's Last Theorem In the statement it requires $a,b,c$ to be positive integers. Is that correct? I always took it to be no solutions in non-zero integers. ...
-2
votes
1answer
280 views

Is this attempted proof of ABC conjecture correct [closed]

This mathematician claims that he has tackled ABC conjecture! He uses induction and simple inequalities to achieve the result. Is this some serious stuff or is there a basic flaw in the reasoning?
3
votes
1answer
64 views

When is $x^2 - 75 y^2 = 0$ in $\mathbb{Z}_p$ solvable?

Exercise: For which prime numbers does the equation $x^2 - 75 y^2 = 0$ have non-trivial solution in the $p$-adic integers $\mathbb{Z}_p$? For $p\neq 5$, the non-trivial solvability of the ...
0
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2answers
24 views

Two real numbers which belong to distinct classes in the quotient group $\mathbb R/\mathbb Z$.

Let $x,y$ two real numbers. What does mean, in "pratical terms", that "$x,y$ belong to distinct classes in the quotient group $\mathbb R/\mathbb Z$"? Maybe that their difference $x-y$ isn't an integer ...
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2answers
38 views

Relevance of prime being divisble by $4k+1$ in proof that 'There are infinitely many primes of the shape $4k+3$'

Show that there are infinitely many primes of the shape $4k+3$ Proof: $1)$ Suppose that there are only finitely many such primes, say $p_1,...p_n$. $2)$ Consider the integer $Q=4p_1...p_n-1$ $3)$ ...
1
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1answer
22 views

Legendre Symbol confusion

How do I know that $\sum\limits_{x=1}^{p-1}\left(\frac{x}{p}\right) = 0$ (where $\left(\frac{a}{b}\right)$ is the Legendre symbol and $p$ is an odd prime)? I know that there must be an even ...
1
vote
1answer
48 views

Proving that an element generates $\mathcal{O}_K^*/ (\mathcal{O}_K^*)^3$

Let $K = \mathbb{Q}[x] / \langle x^3 + 2x^2 + 6x + 6\rangle$. The polynomial has a single real root and its discriminant is $-588$. Let $\alpha$ be the image of $x$ in $K$. Then how would I show that ...