Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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1
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1answer
218 views

How to prove $\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$?

I have to prove for $n \in \mathbb{N}>1$ with $n=\prod \limits_{i=1}^r p_i^{e_i}$. $f$ is a multiplicative function with $f(1)=1$: $$\sum_{ d \mid n} \mu(d)f(d)=\prod_{i=1}^r (1-f(p_i))$$ How I ...
4
votes
1answer
623 views

Why is $n\choose k$ periodic modulo $p$ with period $p^e$?

Given some integer $k$, define the sequence $a_n={n\choose k}$. Claim: $a_n$ is periodic modulo a prime $p$ with the period being the least power $p^e$ of $p$ such that $k<p^e$. In other words, ...
0
votes
1answer
176 views

A reference and an explanation needed?

In my previous question I was asking for a method to construct a global point if we have local points with us which is here, but I got an answer, it didn't serve the entire purpose, but later on due ...
2
votes
2answers
254 views

A Problem on the Möbius Function

$$\sum_{d^2|n}{\mu(d)}=|\mu(n)|$$ I don't know how to prove it, and could you give me some suggestions?
-1
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2answers
126 views

Proof of even like powers?

Can someone show me the proof that difference of like even powers of any two numbers is divisible by the sum of the bases?
2
votes
1answer
136 views

A property of different in Dedekind domains

Let $A \subseteq B$ be a finite extension of Dedekind domains such that the extension $K \subseteq L$ of their quotient fields is separable. Let $\mathfrak{p}$ be a maximal ideal of $A$ and let ...
4
votes
1answer
1k views

Integers in biquadratic extensions

Where can I find information (at least examples) about factorization of prime ideals in biquadratic extensions of $\mathbb{Q}$. Right now I have no idea how, for example, find factorization of $(2)$ ...
1
vote
1answer
205 views

A Generalization of Cantor's counting theory

This question may be silly to experts, but I am waiting for a response sir. My question is " Is there any existence of generalized Cantor's counting principle ( i.e the theory that decide ...
0
votes
1answer
216 views

How do I use Cipolla's algorithm to compute a square root of $56$ mod $101$

I understand that for the first step, I just have to find an integer $t$ such that $0\leq t\leq 100$ such that $t^2-56$ is not a perfect square mod $101$. I know that to show that $t^2-56$ is not a ...
6
votes
3answers
397 views

Proof that for all distinct primes $p, q$, there exists $n$ so that $p+n$ is prime, but $q+n$ isn't

Imagine two distinct prime numbers $p$ and $q$. Intuitively, I'd say that there is always a natural number n so that $p+n$ is a prime number, but $q+n$ isn't. I was given two hints: for each ...
2
votes
1answer
204 views

Denjoy's probability argument for the Möbius function

How or where could I find the proof of Denjoy's probability argument for the Mertens function $$ M(x) = \sum_{n=1}^x \mu(n) = O(x^{1/2}+e) $$ with $e \to 0$ based on the fact that the Möbius ...
3
votes
3answers
375 views

Calculating $\pi(x)$ , a new idea?

I am asking myself if instead of working with the primes in the calculation of $\pi(x)$ up to $x$, we instead work with the composite numbers and then using a simple subtraction to get $\pi(x)$. After ...
12
votes
1answer
242 views

A local-global problem concerning roots of polynomials

Let $f(x)$ be a polynomial with integer coefficients, irreducible over the integers. Suppose that for all primes $p$, $f$ has a zero in the field $\mathbb{Q}_p(\sqrt{2})$. Here $\mathbb{Q}_p$ denotes ...
1
vote
2answers
200 views

Problems about “necklace”

Here necklace has its common combinatorial sense. If the type of a necklace is the number of things in the necklace, 1100:1100, 0110, 0011, 1001--type 4 then how many necklaces are there of type n, ...
3
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1answer
124 views

Factorization of zeta functions and $L$-functions

I'm rewriting the whole question in a general form, since that's probably easier to answer and it's also easier to spot the actual question. Assume that we have some finite extension $K/F$ of number ...
3
votes
1answer
338 views

Why are Gram points for the Riemann zeta important?

Given the Riemann-Siegel function, why are the Gram points important? I say if we have $S(T)$, the oscillating part of the zeros, then given a Gram point and the imaginary part of the zeros (under the ...
1
vote
1answer
95 views

Nice description of $(1+2\mathbb{Z}_2)^{2^k}$?

For $p\neq 2$ it's easy to prove through the log/exp-correspondence that $$(1+p\mathbb{Z}_p)^{p^k}=1+p^{k+1}\mathbb{Z}_p.$$ This gives an easy way to compute the groups ...
2
votes
1answer
256 views

Decomposition of Tate-Shafarevich group

We all know that Tate-Shafarevich group is defined as $$Ш(E/K)=\mathrm{Ker}(H^1(K,E)\mapsto \prod_{v}H^1(K_v,E))$$ for an abelian variety $A$ defined over a number field $K$, the non-trivial ...
3
votes
3answers
189 views

Proofs from the BOOK: Bertrand's postulate: $\binom{2m+1}{m}\leq 2^{2m}$

I have a very hard proof from "Proofs from the BOOK". It's the section about Bertrand's postulate, page 8: It's about the part, where the author says: $$\binom{2m+1}{m}\leq 2^{2m}$$ because ...
4
votes
1answer
168 views

Compactness theorems of adeles and ideles

I've been reading about adeles and ideles and many authors like Milne and Lang spend some time discussing compactness results related to them. This seemed to me more like a technical point until I ...
0
votes
1answer
192 views

Integral ideals of norm less than the Minkowski Bound

Consider a number field $K$ and suppose I want to find the class group and class number of $K$. One of the first steps is to compute the the Minkowski bound. Suppose our bound is $B$. In all the ...
2
votes
2answers
243 views

Number theory counting sums of squares modulus

This is an extra question from an old examination paper: VI. Let $n>1$ in $\mathbf{Z}$ and let $r(n) = \#\{(a,b)\in \mathbf{Z}^{2};n=a^{2}+b^{2}\}$ Let also ...
12
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2answers
671 views

How are the Tate-Shafarevich group and class group supposed to be cognates?

How can one consider the Tate-Shafarevich group and class group of a field to be analogues? I have heard many authors and even many expository papers saying so, class group as far as I know is ...
3
votes
1answer
220 views

Proofs from the BOOK: Bertrand's postulate: proof $\lfloor \frac{2n}{p^k} \rfloor - 2 \lfloor \frac{n}{p^k} \rfloor= 2$

I have a very hard proof from "Proofs from the BOOK". It's the section about Bertrand's postulate, page 9: It's at the top of the page. We want to know, how often the prime factor p divides ...
5
votes
1answer
192 views

A question about the solutions of $y^2 = x^3 - 4$ for $(x,y) \in \mathbf Z^2$

Here's a question from an old examination paper: Find all $(x,y)$ in $\mathbf{Z}^{2}$ where $y$ is odd and $y^2=x^3-4$. Find all $(x,y)$ in $\mathbf{Z}^{2}$ with $y$ even and $y^2=x^3 -4$. ...
3
votes
2answers
245 views

Proofs from the BOOK: Bertrand's postulate Part 3: $\frac{2}{3}n<p \leq n \rightarrow$ no p divides $\binom{2n}{n}$

I have a very hard proof from "Proofs from the BOOK". It's the section about Bertrand's postulate, page 9: I have to show, that for $\frac{2}{3}n<p \leq n$ there is no p which divides ...
3
votes
0answers
198 views

How to find $\beta$ and $\alpha$?

$\mathbb{P}$ is the prime numbers set. $p \in \mathbb{P}$ $a,b,c \in \mathbb{N}$ $n=a p^b+c$ where $c= n\bmod p$ $b$ is the highest power of $p$ who divides $n-c$ How to find $\beta$ where ...
5
votes
2answers
240 views

Is an algebraic formula for the number of cyclic compositions of n known?

From Wikipedia: In January 2011, it was announced that Ono and Jan Hendrik Bruinier, of the Technische Universität Darmstadt, had developed a finite, algebraic formula determining the value of p(n) ...
6
votes
2answers
196 views

Proving two sequences identical

I found something quite interesting while browsing around the OEIS yesterday. I have no idea how to prove this (I don't even know if it's true in general, but Mathematica tells me that it holds up to ...
2
votes
4answers
250 views

Summing up Problem (Combinations)

This is a part of a bigger problem I was solving. Problem: $N$ is a positive integer. There are $k$ number of other positive integers ($\le N$) In how many ways can you make $N$ by summing up any ...
4
votes
1answer
128 views

Least cardinality of a set of integers

If $S_n$ is a set of positive integers >0 of the least cardinality such that every positive integer less then $n$ can be written as the sum of at most two elements of $S_n$, how precisely can we bound ...
3
votes
2answers
991 views

Number of ways to represent a number from a given set of numbers

I want to know in how many ways can we represent a number $x$ as a sum of numbers from a given set of numbers $\{a_1.a_2,a_3,...\}$. Each number can be taken more than once. For example, if $x=4$ and ...
2
votes
3answers
374 views

Are there infinitely many primes of the form $k\cdot 2^n +1$?

Let's observe following matrix with an infinite number of elements : Elements of the main diagonal are of the form $n\cdot2^n+1$ . These numbers are known as Cullen numbers . It is an open question ...
7
votes
1answer
440 views

Reference request: $L$-series and $\zeta$-functions

Does anyone know a good book, lecture note, article etc. on $L$-series (Dirichlet, Hecke, Artin) and $\zeta$-functions in number theory? I'm especially interested in material explaining the following: ...
3
votes
1answer
265 views

If two primes differ by $n$, then infinitely many primes differ by $n$

A proof I'm writing rests on something I can't prove, probably beyond my knowledge, but it seems right: For any two primes $p_k, p_l$ (not necessarily consecutive) such that the distance between ...
2
votes
1answer
125 views

A problem about $\varphi(n)$

Let $\omega$ be the number of distinct primes dividing n, then $$\varphi(n)\geq n\prod_{k=2}^{\omega(n)+1}\left(1-\frac{1}{k}\right)=\frac{n}{\omega(n)+1}$$ Also $2^{\omega(n)}\leq \tau(n) \leq ...
4
votes
2answers
805 views

The Sum of the Odd Divisors of n

The sum of the odd divisors of n is $-\sum_{d|n}(-1)^{n/d}d$, and if n is even, then $$\sum_{d|n}(-1)^{n/d}d=2\sigma(n/2)-\sigma(n)$$ Could you give me some hints on that?
1
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1answer
222 views

Solution of Diophantine equation

I have seen $3^x$ + $3^y$ = $6^z$ and $4^x$ + $18^y$ = $22^z$ on lecture series of Prof. Gandhi. In my own study, I have constructed the following theorem (I am not sure about solvability) and I am ...
4
votes
1answer
208 views

Is $k^2+k+1$ prime for infinitely many values of $k$?

Let's define an infinite sequence of positive integers as : $a_n=k^2+(2n-1)k+2n-1 $ , where $ k,n \in \mathbf{Z^{+}}$ Suppose that one can prove that this sequence contains infinitely many ...
1
vote
0answers
129 views

Binary sequences in primes

Is anything known about these problems? If we make a string S of 0's and 1's with 1 in n'th position if the the nth prime $p_n$ is of the form $1+m 2^{9^{9^{9^{9}}}}$, else 0, does every finite string ...
4
votes
2answers
557 views

What is importance of the Bunyakovsky conjecture?

Bunuyakovsky conjecture states that: An irreducible polynomial $f(x)$ of degree two or higher with integer coefficients and property that $\gcd(f(1),f(2),......)=1$ generates for natural ...
11
votes
2answers
236 views

Special Cases of Quadratic Reciprocity and Counting Fixed Points

The general theorem is: for all odd, distinct primes $p, q$, the following holds: $$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$ I've discovered the ...
3
votes
2answers
206 views

Elementary Sums involving the Kronecker Symbol

Let $\chi_{k}(n)$ denote the Kronecker symbol $(k|n)$ and $\mathbb{Z}_{k}^{\times}$ the group of units modulo $|k|$. Under which assumptions on $k \in \mathbb{Z} \setminus \{0 \}$ can one conclude the ...
21
votes
6answers
6k views

prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer

Prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer. For clarity: the denominator is the only part being squared. My thought process: The numerator is the product of the first n even ...
2
votes
2answers
412 views

If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$

How to prove that: If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$ This statement is generalization of the statement from my previous question. I have checked for many $(a,b)$ ...
1
vote
2answers
312 views

If $\gcd(a,b)=1$ , and $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$?

How to prove that: $\gcd(a,b)=1 \Rightarrow \gcd(2^{a}+1,2^{b}+1)=1$ ,where $a$ is even and $b$ is odd natural number For example: $\gcd(2^8+1,2^{13}+1)=1 , \gcd(2^{64}+1,2^{73}+1)=1$ I know ...
3
votes
1answer
119 views

Show that the p-adic integers are the completion of Z with respect to the p-adic metric

Show that $\mathbb{Z}_p$ = $\varprojlim_n \mathbb{Z}/p^n$ is the completion of $\mathbb{Z}$ with respect to the metric $(x, y) \rightarrow \|x-y\|_p$, i.e, the p-adic metric. I've tried doing this ...
4
votes
2answers
141 views

For $p$ prime, $p = a^2 + b^2$, why must $a$ or $b$ be a square?

If p prime and if $p = 1 \pmod{4}$, then $p = a^2 + b^2$; why must $a$ or $b$ be a square mod $p$?
1
vote
1answer
46 views

Let $n=pg$, then we get $p-1 \mid n-1$ because there are multiplicative groups of integers modulo n

I have one more question: Let $n=pq$ with $p,q \in \mathbb{P}$, then we have $p-1 \mid n-1$ and $q-1 \mid n-1$. I don't understand the reason the author tells me: "because there are multiplicative ...
4
votes
2answers
309 views

How to prove a fact about the sum of three squares?

How would I go about proving the following? If $a$, $b$, $c$, $n$ are positive integers, then $a^2+b^2+c^2 \neq 2^nabc$ I tried doing something similar to the proof for Adrien-Marie Legendre's ...