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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Coprime multiplicative orders modulo infinitely many primes

Is it true that there are infinitely many primes $p$ such that the multiplicative orders of $2$ and $3$ are coprime $\pmod{p}$? By this I mean their order in $(\mathbb{Z}/p\mathbb{Z})^*$. If the ...
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102 views

About solutions to $x^2+y^2+z^2=8n+3,n\in \mathbb N$

As we know that $x^2+y^2+z^2=8n+3,(n\in \mathbb N)\tag{1}$ has integer solutions $x,y,z\in \mathbb N.$ If $k\in \mathbb N$ has at least one prime factor which is $\equiv 3 \mod 4,$ then we call $k$ ...
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502 views

Natural extension of the divisor function $\sigma$ to the complex integers $\mathbb{Z}[i]$

The divisor function $\sigma$ of a positive integer $n$ is defined as the sum of the positive divisors of $n$. It turns out that this definition is equivalent to ...
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887 views

The prime number theorem and the nth prime

This is a much clearer restatement of an earlier question. In section 1.8 of Hardy & Wright, An Introduction to the Theory of Numbers, it is proved that the function inverse to $ x ⁄ \log⁡ x$ is ...
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72 views

What are the densities of branches of the euclidean tree?

The Euclidean algorithm shows how all coprime pairs of positive integers can be uniquely obtained from the pair $(1,1)$ by applying the two operations $(a,b) \to (a+b,b)$ and $(a,b) \to (a,a+b)$. (or ...
7
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823 views

How to use Hardy and Wright's text and what corresponding exercises/problem books can I do?

I have just started out with Hardy and Wright's An Introduction to the Theory of Numbers today. I find the lack of exercises in the book as a departure from the style of the textbooks we are so ...
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295 views

Sums of Dirichlet-Characters over prime numbers (part 2)

This is kind of related to my previous question that was poorly stated because of misreading my own notes that I have taken on the papers I am currently reading, so no surprise that it eventually ...
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164 views

closure of units of number fields in the finite idele topology

Let $K$ be a number field. Denote by $\mathcal O _K^\times$ its rings of units and by $\mathcal O _{K,+} ^\times$ its ring of totally positive units. Further let us denote by $\mathbb A ...
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44 views

A counterexample to $x^n + y^n = h^2 + nf^2$ implies $x + y = h'^2 + nf'^2$ in the integers

The Wikipedia page for Sophie Germain contains the following: In the same 1807 letter, Sophie claimed that if $x^n + y^n$ is of the form $h^2 + nf^2$, then $x + y$ is also of that form. Gauss ...
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91 views

Twin-prime sieve

My question concerns the following sieve (call it S), which was an exercise in applying some elementary aspects of Brun's sieve while reading Halberstam's text. Using the Chinese Remainder theorem ...
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105 views

Discrete valuation fields and representation as power series

Let $(K,v)$ be a discrete valuation field ($v$ is surjective). Let $\mathcal O$ be the ring of integers of $v$ and moreover let $\mathfrak p$ be the unique maximal ideal of $\mathcal O$. Then we have ...
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209 views

A combinatoric $gcd$ problem

Let $Q(L)$ be the number of pairs of numbers $m , n$ such that $gcd(m,n) = 1$ and $m$ and $n$ are of different pairity, where $m$ is even and $n$ is odd, and $m^2 + n^2$ $\le$ $ L$. $$Q(L) = ...
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113 views

Solving (n+1)(n+2)…(n+k)−k = x^2

Let $n$ and $k$ be positive integers. Need to find all pairs of $(n,k)$ such that $$(n+1)(n+2) \cdots (n+k)−k = x^2,$$ where $x^2$ is a perfect square.
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122 views

Goldbach for certain classes of $n$

The Wiki article on the Goldbach conjecture (where $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$) states that In 1975, Hugh Montgomery and Robert Charles ...
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108 views

For a given integer $n$, how many primes $p_1,p_2 \leq n$ such that $\tau(p_1-1)=\tau(p_2-1)$

This is a curiosity question. Let $N$ be positive integer, I just want to know how many (an approximation) pair of primes $(p_1,p_2)$ that are less than $n$ and verify the following identity: ...
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137 views

Can a sum of three fifth power of integers be 8?

By congruence computation we get that $n= a^5+b^5+c^5$ implies $n \not \equiv 4,5,6,7 \pmod{11} $ (with $a,b,c \in \mathbb{Z}$) For $a,b,c \in \{-100,-99, \dots , 99, 100\}$, the set of integers ...
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85 views

Elementary proof that finite sums of square roots of primes is irrational

It is relatively easy to show that if $p_1$, $p_2$ and $p_3$ are distinct primes then $\sqrt{p_1}+\sqrt{p_2}$ and $\sqrt{p_1}+\sqrt{p_2}+\sqrt{p_3}$ are irrational, but the only proof I can find that ...
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64 views

Divisibility sequence resulting in limit with pi

Consider the following sequence of operations : Start with a natural number $n$ and then round it up to the closest multiple of $n-1$ .Then round up this new number to the closest multiple of ...
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63 views

Need help with technical detail in Dedekind's 1872 paper

I found the restatement of Dedekind's construction of real numbers in Rudin quite confusing. As a German speaker, I started to read the paper by Dedekind (1872), which is very beautiful and gives a ...
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132 views

From $\prod_{d\mid n}d=n^{\sigma_0(n)/2}$ to $n!=\operatorname{lcm}(1,\ldots,n)^{e(n)}$, where $\sigma_0(n)$ is the number of divisors

We know that $$\prod_{d\mid n}d=n^{\sigma_{0}(n)/2}$$ for every integer $n\geq 1$, where $\sigma_{0}(n)$ is the number of positive divisors of $n$, see for example [1] (exercise 10, page 47). And for ...
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67 views

Serre's surjective theorem importance.

I'm studying Serre's paper in wich he shows the following theorem: Let K be a number field, $E$ an elliptic curves over K without CM. Then the representation $$\rho_{\ell}:\mathrm{Gal}(\bar ...
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47 views

For all Dirichlet series, is $a_n$ unique to $f(s)$?

For any Dirichlet series, $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ is the function, $a_n$, always unique to $f(s)$? In other words, is it possible to show that $a_n$ is the only function that will ...
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87 views

What primes were “pending” at the time of Wiles's proof of FLT?

I would like to know what instances of Fermat's Last Theorem were pending at the time of Wiles's proof. More specifically: what families of irregular primes had been discarded as possible ...
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214 views

One-to-one correspondance between zeta zeros and the prime powers?

I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2 $ in the ...
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106 views

A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
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147 views

Does $\pi$ contain infinitely many “zeros” in its decimal expansion?

Some number doesn't contain $"7"$ in its decimal expansion. For example Liouville's constant $$L=\sum_{n=1}^\infty\frac{1}{10^{n!}}=0.11000100....$$ contains only $0$ and $1$. It is well-known ...
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220 views

Relatives of Heegner numbers?

It is well known that Euler's lucky numbers are related to the Heegner numbers, where \begin{align} &n^2+n+p\\ \end{align} gives primes for $n=0,\dots,p-2$ if and only if its discriminant ...
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86 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 ...
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97 views

Representing of natural number

Let $n$ be a given natural number with $n>1$, please prove that each natural $x>=2n$ can be represented as $$x=a_1^n+a_2^n+\cdots+a_n^n-(b_1^n+\cdots+b_{n-1}^n)$$ where ...
6
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77 views

To solve for $x,y,n$ in non-negative integers , $\dfrac{x!+y!}{n!}=p^n$ , $p$ a given prime

Let $p$ be a given prime , then how do we find non-negative integers $(x,y,n)$ $\space$ , such that $\dfrac{x!+y!}{n!}=p^n$ ?
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112 views

Involutions of the second type in a division algebra

I'm trying to figure out some details about involutions of division algebra, thought maybe someone here might have a better insight. Let $k$ be a $p$-adic or number field, and let ...
6
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142 views

There exist infinitely many positive integers $k>r!$, such that for any $j$ with $r!<j<k$ we have $\prod_{i=0}^{r-1}(j-i)\nmid\prod_{i=0}^{r-1}(k-i)$

Question: Let $r$ be a postive integer. Show that there exist infinitely many positive integers $k$ satisfying $k>r!$,such that for any positive integer $j$ satisfying $r!<j<k$ we have ...
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136 views

Are there infinitely many non-negative integers not covered by one of these 7 polynomials?

Consider the following polynomials: $$ \begin{align} f_1(n, m) &= 30nm + 23n + 7m + 5 \\ f_2(n, m) &= 30nm + 17n + 13m + 7 \\ f_3(n, m) &= 30nm + 23n + 11m + 8 \\ f_4(n, m) &= ...
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120 views

Ramification index of infinite primes

I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an ...
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131 views

How many solutions are there to $2^a + 3^b = 2^c + 3^d$?

Are there only finite quadruples of non-negative integers $(a,b,c,d)$ that satisfy the following equation: $$2^a + 3^b = 2^c + 3^d \quad ?$$ with $a \neq c$. I found these: $5 = 2^2 + 3^0 = 2^1 + ...
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70 views

How to prove $W(k)$ is a complete discrete valuation ring?

I am trying to prove the fact that the ring of Witt Vectors $W\left(k\right)$ is a complete discrete valuation ring, where $k$ is a perfect field of characteristic $p$ , but I'm stuck. Theorem 2 ...
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150 views

even square numbers represented as two primes added together

Can every even square number be written as the sum of 2 prime numbers? How do you prove the result?
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129 views

A 9 dimensional number system?

Given $x$ and $y$ as sums of nine cubes. When we multiply them, $xy$ may again be written as a sum of nine cubes, due to Wieferich and/or Kempner. Does this invent a 9 dimensional number system, just ...
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159 views

Twin Prime Powers

What are all the possible triplets of numbers $a$, $b$, $c$ such that $a+2=b$, $a+4=c$, and all $3$ are prime powers (where one must be a power of $3$)? I'm aware of the cases for when they are ...
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201 views

expressing $\log(\left \lfloor x \right \rfloor!)$ in terms of zeta-zeros

let $\psi(x)$ be the second Chebyshev Function. By the definition of this summatory function, and the fundamental theorem of arithmetic, we have the identity: $$\log(\left \lfloor x \right ...
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266 views

How Ramanujan find this formula

I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...
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125 views

A formula to calculate summations over all divisors of a fixed integer

I don't know much about number theory, it seems this summation might involve some facts from number theory. Could you give me some idea of doing it? Thank you very much. The summation is $$ ...
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476 views

Sum of reciprocal of primes in arithmetic progression

In http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf on page 6 (top) the author states that: $$ \sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} = \frac{\log \log x}{\phi(l)} + O \left ( \frac{\log ...
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83 views

Statement about Woodal primes.

A Woodal number is an integer of the form $n 2^{n}-1$. A Woodal prime is an integer that is both a prime and a Woodal number. Let $p$ be a prime of the form 1 mod 4. Then $p 2^{p} -1$ is never a ( ...
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341 views

Taxicab numbers.

I think most people know these numbers. Find $x,\ y,\ z,\ w$ such that $x^3 + y^3 = z^3 + w^3$ and $x,\ y,\ z,\ w$ are not equal to each other. The first is $1729$. I'm trying to figure out if ...
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156 views

Density of products of a certain set of primes

I have a set S of prime numbers and I would like to find the size (in some sense, ideally some nice asymptotic expression) of the set of positive integers which are the product of with all prime ...
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0answers
133 views

About the isomorphism $\operatorname{Br}(\mathbb{Q}_p)\cong \mathbb{Q}/\mathbb{Z}$

I've been reading the section about Brauer groups in Introduction to Modern Number Theory, and I couldn't quite understand how this isomorphism is defined. We start with a central simple algebra $A$, ...
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154 views

Number of primefactors in $ f(n,W) = \prod_{k=1}^W (p_k^n -1) \text{ where } p_k=Prime(k) $

I'm reviving an old fiddling, although I do not yet really see its benefit. Beginning with the eulerproduct for the zeta-function in the representation $\small \zeta(n)=\prod {1\over 1-p^{-n} } = ...
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427 views

Ramanujan Series

I have the following question involving the series of $1/\pi^3$: Can we find such expansions by using the one for $1/\pi^3$ with $1/\pi^4$ or $1/\pi^n$ etc? Note that $$\frac{1}{32}\sum_{n=0}^\infty ...
6
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347 views

Certain permutations of the set of all Pythagorean triples

The fact that the set of all primitive Pythagorean triples naturally has the structure of a ternary rooted tree may have first been published in 1970: http://www.jstor.org/stable/3613860 I learned ...