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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3
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0answers
33 views

Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of ...
7
votes
0answers
82 views

prove that $x^2 + 5 =y^3$ has no solutions for $x,\ y \in \mathbb{Z}$ [duplicate]

So the question is completely stated by the title. My own thoughts: I can prove that $x^2 + 1 = y^3$ has no solutions for $x,y \in \mathbb{Z}$ by using the factorization: $$ y^3 = (x-i)(x+i) $$ in ...
0
votes
1answer
90 views

Integer solution to the equation

Does there exists an integer solution (for every integer $m\geq 1$) for the following equation? $$x_1x_2...x_n+(2y+1)z+y=4m+3$$ where, $1\leq x_1\leq x_2\leq...\leq x_n\leq l$,$0\leq y \leq ...
0
votes
3answers
18 views

system of conguences

I am trying to figure out how to solve: Find $x, y \in \mathbb{Z}$ such that $$2x+y\equiv 4\pmod{17}$$ and $$5x-5y\equiv 9\pmod{17}.$$ I know this means $2x+y-4$ and $5x-5y-9$ are multiples of $17$, ...
2
votes
1answer
19 views

Looking for different proofs for $-1$ is a quadratic residue of primes of the form $4k+1$ and related facts

Suppose $p$ is a prime of the form $4k+1$ , then $4|p-1=|\mathbb Z_p^*|$ , as $\mathbb Z^*_p$ is a cyclic group , so there is $\bar x \in \mathbb Z_p^*$ such that $o(\bar x)=4$ , then $o(\bar x^2)=2$ ...
0
votes
0answers
22 views

Question related to image of $[1,N]^n$ under a linear tranformation

I am reading an article and I am a bit confused about the following passage. I would appreciate any clarification. It goes as follows: Let $\bar{F}$ be a collection of $r$ linearly independent ...
0
votes
1answer
49 views

Does this compound interest problem coincide to the value of e by coincidence?

An account starts with €$1.00$ and pays $100\%$ interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be €$2.00$ . What happens if the ...
2
votes
1answer
48 views

$3 X^3 + 4 Y^3 + 5 Z^3$ has roots in all $\mathbb{Q}_p$ and $\mathbb{R}$ but not in $\mathbb{Q}$

This is an exercise in my textbook in a chapter about the Hasse-Minkowski-theorem: Show that the polynomial $3 X^3 + 4 Y^3 + 5 Z^3$ has a non-trivial root in $\mathbb{R}$ and all $\mathbb{Q}_p$. ...
0
votes
0answers
29 views

Determine all $n \in \mathbb{N}$ such that $GCD(n,48)=6$, $14|n$ and $|Div^+(n)|=12$.

Determine all $n \in \mathbb{N}$ such that $\gcd(n,48)=6$, $14|n$ and $|Div^+(n)|=12$. What I did: $14|n$ then $2|n$ and $7|n$ so $n=2\cdot7\cdot q$, $q \in \mathbb{Z}$. Then $6|n$ implies $2|n$ and ...
4
votes
0answers
55 views

Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?

Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$): $$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) ...
-1
votes
2answers
70 views

Squares numbers into squares

Solve the two related questions below in which lowercase letters are digits in base $10$, $a > 0$ and $N\in\Bbb N$. Find the values of $N$ in $(1)$ and prove or deny $(2):$ $$ ...
0
votes
0answers
24 views

the asymptotic approximation of a sum

$p_{n}$ and $p_{j}$ are two primes with $p_{n}<p_{j}$ where the $n$ and $j$ denotes the $n$th and the $j$th prime. I have this sum $$\sum \limits^{k=\frac{b-p^{2}_{n}p_{j}}{2p_{n}p_{j}} ...
1
vote
2answers
50 views

Are there arbitratily long runs of consecutive integers n that are NOT of the form $n = p^k$ or $n = 2p^k$ for some $k>0$ and $p$ an odd prime?

I know that there are arbitrarily long runs of consecutive non-primes. I am interested in this question because an integer $n>4$ has a primitive root if and only if it has the form $n = p^k$ or ...
0
votes
0answers
18 views

Generate a function that shuffles a number withing a given range which is reproducible

Lets say I have an array of numbers $1 2 3 4 5 6 7$. I want to shuffle these numbers in some order , $7 5 4 3 1 26$ . However , it should be revesrible. That is given the second array I must be able ...
0
votes
1answer
21 views

Exercise 2.13 from a computational introduction to number theory and algebra

This is an exercise from V. Shoup. A computational introduction to number theory and algebra. Let $p=2, e=3, a=b=1, c = 0$, then $p^{2e} = 64, z\in \{0,1,2,\cdots,63\}$, the conclusion is, there ...
1
vote
1answer
56 views

Show that $\limsup_{x \to \infty} \frac{\pi(x)}{x/ \log x} \geqslant 1. $

Show that $$\displaystyle\limsup_{x \to \infty} \dfrac{\pi(x)}{x/ \log x} \geqslant 1. $$ I've seen $\displaystyle\lim_{x \to \infty}$ operator, but I haven't seen $\displaystyle\limsup_{x \to ...
1
vote
2answers
48 views

What is the value of $1^2 + 2^2 + 3^2 + \cdots + (p-1)^2\pmod{p}$?

What is the value of $1^2 + 2^2 + 3^2 + \cdots + (p-1)^2\pmod{p}$? Let's try a several primes greater than 3... If $p=5$, then we have $1^2 + 2^2 + 3^2 + 4^2 = 30$, so that $30\pmod{5} = 0$ If ...
6
votes
2answers
87 views

Prove or disprove : $a^3\mid b^2 \Rightarrow a\mid b$

I think it's true, because I can't see counterexamples. Here's a proof that I am not sure of: Let $p_1,p_2,\ldots, p_n$ be the prime factors of $a$ or $b$ \begin{eqnarray} a&=& ...
1
vote
0answers
27 views

Prove that $a_{i,1}x_1 +a_{i,2}x_2 +···+a_{i,n}x_n ≤c_i, 1≤i≤n $ are all satisfied by a nonzero $n-tuple$ of integers.

My setting is that $c_1, · · · c_n$ are positive real numbers, and $A = [a_{i,j} ]$ is an $n × n$ non-singular matrix. Assume that $c_1 · · · c_n > | det(A)|.$ I want to prove that the n-linear ...
4
votes
3answers
60 views

Show that $\sum\limits_{p \leqslant x}1/p = \frac{\pi(x)}{x} + \int_2^x \frac{\pi(u)}{u^2} du.$

Show that $$\displaystyle\sum\limits_{p \leqslant x}1/p = \dfrac{\pi(x)}{x} + \int_2^x \dfrac{\pi(u)}{u^2} du.$$ In the equation above, $\pi(x)$ denotes the prime counting function. To get ...
3
votes
1answer
97 views

Product of Stirling Numbers of the first kind

I have been messing around with coefficients of various polynomials and was wondering if there was a way to reduce the following stuff. Let polynomial, ...
0
votes
0answers
35 views

Prime - number theory [duplicate]

Why is the digit 1 is not a prime number? 1 can be devided by 1 and itself. I think it's because we can express like 1= 1x1x1 ... is it true or not?
3
votes
0answers
42 views

Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree ...
2
votes
1answer
51 views

Why is $\mathbb{Q}_\infty = \mathbb{R}$?

Why, in the context of p-adic numbers, do we have the convention $$\mathbb{Q}_\infty = \mathbb{R} \quad$$ ? It must have something to do with the generalization of the Legendre-symbol for ...
1
vote
4answers
126 views

If $a$ and $b$ are positive integers and $4ab-1 \mid 4a^2-1$ then $ a=b$.

Prove that if $a$ and $b$ are positive integers and $$(4ab-1) \mid (4a^2-1)$$ then $a=b$. I am stuck with question, no idea. Is there any way to prove this using Polynomial Division Algorithm? Would ...
1
vote
0answers
34 views

Sums of triangles

If $P$ is any odd prime, is there a proof that working Mod P, each number from $0$ to $P-1$ except $\frac{(P^{2} -1)}{4}$ can be formed as the sum of the triangle of a number <(P+1)/2 and the ...
-2
votes
2answers
33 views

How do you show that$ ∏j ≡1 $(mod p) where j is $1 \le j\le p-1$ and $\frac{j}{p}=1$ [on hold]

Also, $P$ is a prime of the form $4k+3$ and $k$ is an element of natural numbers including $0$.($\frac{j}{p}$) denotes a legendre symbol.
-1
votes
0answers
21 views

is find magic square from even degree?

please tell me is there any journal that find magic square to degree of even? I find a good way to solve it but not for all degree actually for a particular degree of it.for example degree ...
1
vote
2answers
13 views

Prove the order of the group homomorphism of an element divides the order of the element.

Let $\phi : G \rightarrow H$ be a group homomorphism. Prove $\forall g \in G$, the order of $\phi(g)$ divides $g$. I've gotten to the point where I've shown that if, $ord(\phi(g)) < ord(g)$ then ...
2
votes
1answer
27 views

How many pairs of polynomials $(U,V)\in \Bbb Z[x]^2$ such that $P=U^2+V^2$ for a given polynomial with integer coefficients?

This question is no more than curiosity question. For integers we know that a positive integer $n$ is a sum of two squares if and only if for any prime $p$ such that $p\equiv 3 \mod 4$ we have ...
4
votes
0answers
81 views

Solve $(x+1)^n-x^n=p^m$ in positive integers

Solve in positive integers: $$(x+1)^n-x^n=p^m$$ $p$ is prime, $n\ge 2$. Seemingly Zsigmondy's Theorem and LTE won't work here. Though you can tell (as suggested by user barto), using ...
1
vote
0answers
21 views

Power series in p-adic integers

How can we show that for $x \in \mathbb{Z}_p$, $\log_p(1+x)$ converges in $\mathbb{Z}_p$ when $|x|_p < 1$? To clarify, $\log_p(1+x)$ is the power series: $$\sum_{n=1}^\infty ...
0
votes
1answer
45 views

Number Theory - Order of $p^2$ [duplicate]

How can you show that if $p$ is prime then the numbers with maximum possible order modulo $p^2$ is $\phi(\phi(p^2))$. I tried finding order(a) modulo 9, and obtained the following: $1$ is $1$, $2$ is ...
2
votes
1answer
50 views

What is behind these series?

I just found out (I am an amateur) that if I have the following series I get the following answers for the a nth number . (each series is the sum of the previous one) $$1 ,1 , 1, \dots, 1 $$ ...
3
votes
1answer
35 views

Primes of the form $x^2 + 3y^2$

Im trying to prove that a prime $p\neq 3$ is of the form $p=x^2 + 3y^2$ if $p \equiv 1 \pmod{3}$. I have think in a prove as follows: As we know that $-3$ is a quadratic residue mod p, we know that ...
0
votes
1answer
21 views

Why $n^{d(n)/2}$ is not getting satisfied?

Respected all. We know that the product of all positive divisors of $n\in \mathbb N$ is $n^{d(n)/2}$ where $d(n)$ is the number of positive divisors on $n$. What will happen if $d(n)=odd$ say we ...
2
votes
1answer
19 views

Discriminant of n algebraic numbers equals $0$ iff the algebraic numbers linearly dependent

Let $K \subset L$ be two number fields with $[L:K] = n$. Let $\{\alpha_i:1 \leq i \leq n\} \subset L$. Then $\operatorname{disc}(\alpha_1 \dots \alpha_n) = 0 \iff \alpha_i$ are linearly dependent ...
0
votes
1answer
16 views

Showing Zp is isomorphic to the completion of Z, when Z is considered as a p-adic metric space

Question: Show that Zp is isomorphic to the p-adic completion of Z; that is, the completion of Z when Z is considered a metric space via the p-adic metric. I'm stuck. If we take an element a in Zp, ...
2
votes
0answers
30 views

Show that the Euclidean algorithm works for Gaussian integers. [closed]

Here is a question from introductory level number theory at college: Prove or show that the Euclidean algorithm works for Gaussian integers. Thank you very much for your valuable time and help!
2
votes
0answers
29 views

Poles and zeroes

If $f$ be the function defined by $$f(x)=2sin\frac{x}{2}\prod_{k=1}^{\infty}\frac{(1-e^{ix}q^k)(1-e^{-ix}q^k)}{(1-q^k)^{2}}$$ where $q = e^{2\pi it}$ $h(x)=\frac{f'(x)}{f(x)}; \quad \quad ...
-1
votes
0answers
27 views

Prime Factorization of 6

What would be the prime factorization of 6 in $Q[√−1]$? Can I generalize this to other numbers as well or no? Can someone please help me here?
1
vote
1answer
25 views

Quadratic Prime integer norm is not prime

What would be an example of a quadratic integer in $Q[√−1]$ which is prime, but whose norm is not prime?
3
votes
1answer
38 views

What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$?

Theorem #114 in Hardy and Wright says if $p = 4k+3$ then $$ \left[\frac{1}{2}(p-1)\right]! \equiv (-1)^\nu \mod p$$ where $\nu = \# \{ \text{non residues mod } p\text{ less than }p/2\}$. Is ...
2
votes
2answers
29 views

Computing the volume of a fundamental domain of a lattice

Suppose I have $n$ linearly independent vectors in $\mathbb{R}^m$, say $v_1, .., v_n$. Then $v_1,..., v_n$ form a lattice $\Lambda$ inside a subspace $V$ = $\mathbb{R}v_1 + ... + \mathbb{R}v_n ...
0
votes
1answer
33 views

Prove that if p is an odd prime and p does not divide $ac$, then $ax^2+bx+c \equiv 0$ mod $p$ and $cx^2+bx+a \equiv 0$ mod $p$

Prove that if p is an odd prime and p does not divide $ac$, then $ax^2+bx+c \equiv 0$ mod $p$ and $cx^2+bx+a \equiv 0$ mod $p$ have the same number of solutions. Any help would be appreciated.
1
vote
1answer
28 views

Why is determinant called volume of the fundamental parallelepiped in geometry of numbers?

Let $v_1, ..., v_n$ be $n$ linearly independent vectors in $\mathbb{R}^n$. Then they form a lattice $\Lambda \subseteq \mathbb{R}^n$ and the volume of the fundamental domain is $|\det A|$, where $A$ ...
1
vote
1answer
25 views

Prove that $z^2 \equiv ab$ mod $p$ is solvable if and only if both or neither of $x^2 \equiv b$ mod $p$ are solvable.

Suppose the $p$ is an odd prime not dividing $ab$. Prove that $z^2 \equiv ab$ mod $p$ is solvable if and only if both or neither of $x^2 \equiv b$ mod $p$ are solvable. I have no idea how to prove ...
1
vote
2answers
59 views

Diophantine equation resembling FLT

I was wondering if the equation $x^p+y^p=2z^p$ has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular ...
2
votes
1answer
24 views

Finite field extensions - $K(\alpha)$

So I am currently studying Algebraic Number theory and a theorem in the Book states the following: Let $L/K$ be a field extension. Then $\alpha \in L$ is algebraic over $K$ if and only if there is ...
-2
votes
1answer
68 views

Cantor's Diagonal: Why not a 1-2 Correspondence between the Naturals and Reals?

Hopefully I'm following Cantor's Diagonal Argument with a minimum of distortion and omission: We start from an enumeration T of all infinite binary sequences. We then construct a list S of elements ...