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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40 views

Could a determinstic primality test specialized to this form of prime exist?

Is it possible there could be an "efficient" deterministic primality test for prime numbers of the form $$(2^n + 1)^2 - 2$$ or $$(2^n - 1)^2 - 2$$ in the same vein as the Lucas-Lehmer test for ...
2
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0answers
69 views

Unique decomposition of $c$ sums of products of $k$ numbers greater than 1, allowing duplicates?

This question differs from Unique decomposition of $c$ sums of products of $k$ prime numbers, allowing duplicates? in that prime number restriction is changed to any number greater than 1. Suppose ...
2
votes
1answer
50 views

The criteria for two abelian extensions to be embedded

Learning class field theory I found this theorem, but I can't prove it or find the solution. I'll be glad to any help. Let $L$ and $M$ be abelian extensions of $K$. $L \subset M$ if and only if ...
2
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0answers
47 views

Question about paper on Selmer groups

Let $\textrm{Sel}_{n}E$ denote the $n$-Selmer group and $\textrm{Sel}_{p^{\infty}}E = \varinjlim_{n}\textrm{Sel}_{p^{n}}E$. Proposition 5.10 of this paper http://arxiv.org/abs/1304.3971 states that ...
2
votes
1answer
157 views

The set of exponential primes

Consider a set of integers $Q$ such that the set of all positive integers $\mathbb{Z}$ is equivalent to the span of ever possible power tower $$a_1^{a_2^{\ldots a_N}}$$ involving $a_i \in Q$. In ...
2
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0answers
37 views

number of solutions eqauation on supersingular elliptic curve

To Frobenius endomorphism on supersingular elliptic curve I want to prove that equation $\pi_q(X) = A$ has 1 solution for any point $A \in E(\bar{\mathbb{F}_q}))$ where $E$ is supersingular. Is it ...
2
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0answers
74 views

Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: ...
2
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0answers
100 views

A function that generates 'alternating' non-trivial zeros of $\zeta(s)$

I am trying to find a function, that assuming RH, generates subsequent non-trivial zeros $\rho_n$ in an alternating way i.e.: $$\frac12+14.134...i,\frac12-21.022...i,\frac12+25.010...i, \dots$$ or ...
2
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0answers
113 views

Representing a fraction as a $p$-adic number

If we have the following $p$-adic number: $$2+3p+5p^2+2p^3+3p^4+5p^5+2p^6+3p^7+5p^8+.....$$ and I am trying to find what rational number this p-adic number represents. I have no idea as to how to go ...
2
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0answers
69 views

An identity related to Chebyshev polynomial

Let $n=2m$ be a positive even integer. I can prove that $$1+\sum_{k=1}^m (-1)^k \frac{n^2(n^2-2^2)\cdots(n^2-(2k-2)^2)}{(2k)!}=(-1)^m$$ using hypergeometric identity ...
2
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0answers
101 views

Representations of $\text{GL}_2(\mathbb{Q})$

Let's say that as a representation theorist I am naively interested in representations of $G(\mathbb{Q})$, where $G$ is an algebraic group defined over $\mathbb{Q}$. For the purposes of this question, ...
2
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1answer
34 views

$d$ to $1$ map, cyclic group and cosets

We know that $\mathbb{F}_p^{\times}$ is a cyclic group. Let $g$ be a prmitive root mod $p$, then the kernel of the map $\varphi:\mathbb{F}_p^{\times}\rightarrow \mathbb{F}_p^{\times}$ defined by ...
2
votes
0answers
52 views

Modular Quadratic System of Equations

I have a system of quadratic equations of two variables to solve in several moduli: $z_0 \equiv (x+k_0)^2-(x+k_0)y \ (mod\ n_0)$ $z_1 \equiv (x+k_1)^2-(x+k_1)y \ (mod\ n_1)$ ...a $z_m \equiv ...
2
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0answers
35 views

$\sum_{i=1}^{k}[\sqrt{ip}]=\frac{p^2-1}{12}$, $p$ is a prime of the form $4k+1$ [duplicate]

$\sum_{i=1}^{k}[\sqrt{ip}]=\frac{p^2-1}{12}$, $p$ is a prime of the form $4k+1$ How to prove this?
2
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0answers
38 views

Is there any relation between the modular inverse of the same integer under different modulus?

I mean, suppose $$ab \equiv 1 \mod{m}$$ $$ac \equiv 1 \mod{n}$$ I wonder if there is any relation between $b$ and $c$? Could we compute one from another? Thanks in advance!
2
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1answer
93 views

Calculate or bound infimum

Let $a_1, \ldots, a_n \in\mathbb R$ and nonnegative let $b\geq1$ and $c\in [0,1]$. Calculate or bound from above $$ \inf \left\{d>0: \sum_{i=1}^n \ln ...
2
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0answers
89 views

Paradox of Infinity? [duplicate]

If a series such as '$a$' below adds to infinity: $a = 1 + 2 + 4 + 8 + 16 + \cdots\to \infty$ Multiplying '$a$' by $2$ yields: $2a = 2 + 4 + 8 + 16 + \cdots\to \infty$ However when I subtract ...
2
votes
1answer
111 views

Is there any known algorithm for factoring the fractional components of a binomial?

For a binomial such as $\binom {15} {6}=\frac{15\times14\times13\times12\times11\times10}{6\times5\times4\times3\times2\times1}$, it seems that it always divides evenly into an integer, and I ...
2
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0answers
88 views

convergence of a sum with zeroes of zeta function

Can it be proved that the sum of this series is smaller than $x$? $$ \sum_{\zeta(a+ib)=0}u_{a,b}(x)\lt x, $$ for all $x$, with $$ ...
2
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0answers
156 views

Integral basis for an ideal

This is something I was curious about from my algebraic number theory class. Given any non-zero ideal $I$ of $A \cap K$ (algebraic integers in $K$ where $[K:\mathbb{Q}] < \infty$), we know it has ...
2
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0answers
148 views

Prime factors of a random number

Let $r$ be a uniformly random integer between $1$ and $N$, for some large enough $N$ (i.e., I'm only interested in the asymptotics). What is the expected largest prime factor of $r$? Is there a good ...
2
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0answers
67 views

If $x \sim U(Z_n^*)$ then $x^2 \pmod n\sim U(QR_n)$?

Define: $Z_n^*=\{x \in Z_n | \operatorname{gcd}(x,n)=1\}$ $QR_n=\{x \in Z_n | \exists r \in Z_n \; s.t. \; r^2 =x\}$ How can I show that $x \sim U(Z_n^*) \implies x^2 \pmod n \sim U(QR_n)$? Thank ...
2
votes
0answers
64 views

Lower bound on diophantine system of inequalities with all but one non-linear constraint

I have a system of $n+1$ diophantine inequalities, in the following form: $$f_{1}(x_1, x_2, \dots, x_n) \geq 0$$ $$f_{2}(x_1, x_2, \dots, x_n) \geq 0$$ $$\vdots$$ $$f_{m}(x_1, x_2, \dots, x_n) ...
2
votes
1answer
488 views

How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
2
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0answers
30 views

Integrable continuations of an arithmetic function.

Suppose we have the arithmetic functions $a_i : \mathbb{N} \rightarrow \mathbb{R}$ and $b_j : \mathbb{N} \rightarrow \mathbb{R}$ such that $a_n = b_n = 1$ if and only if the index $n$ is a square, and ...
2
votes
1answer
170 views

Techniques to prove that there is only one square in a given sequence [duplicate]

What techniques/methods can be used to prove that the sequence produced by $n\cdot (n+1)\cdot (2\cdot n+1)/6$ contains only one square ($4900$) greater than 1? While this particular sequence is an ...
2
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0answers
120 views

Imposing condition of specification of product of $n$ of imaginary numbers on coefficients of imaginary numbers

I asked the same question but with some fatal mistake that makes the question unanswerable - so I decided to delete it and start new. Connecting from The set of numbers that when multiplied do not ...
2
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0answers
110 views

Diophantus again; not to say Pell.

Is there a way to solve the second degree Diophantine equation in two variables $ax^{2} -ny^{2} = b$ $(1)$ where a and b are known and n is a parameter; all solutions x= f(n) and y = f(n) ? For ...
2
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0answers
181 views

Pentagonal-Triangular numbers

Pentagonal Triangular Number is a number which is simultaneously a pentagonal number $P_n$ and triangular number $T_m$ . Such numbers exist when $$ \frac{1}{2}n(3n-1) = \frac{1}{2}m(m+1) $$ This ...
2
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0answers
101 views

I have the following Diophantine equation $x=y^2-z^2$ [duplicate]

Possible Duplicate: $a^2-b^2 = x$ where $a,b,x$ are natural numbers Show that for every positive odd integer $x$ there exist integers $y$ and $z$ such that $(x, y, z)$ is a solution ...
2
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0answers
232 views

Diophantine equations/Diophantine Geometry

I am very knew to this site and I am eagerly waiting for solutions of: (1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
2
votes
1answer
424 views

Evaluate a double infinite summation

$$ H=\sum_m\sum_n'\frac{1}{(m-1+nz)(m+nz)} $$ The summation of $m,n$ is over $\mathbb{Z}$, and skips $(1,0),(0,0)$,and $z\in\mathbb{H}$ (The upper half plane). The series comes from Jean-Pierre ...
2
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0answers
45 views

Finding reduced quadratic numbers and principal ideals

Hello :) I want to compute alle reduced quadratic numbers with discriminat $65$. We call a number $\gamma$ reduced if $\gamma>0$ and $-1<\gamma'<0$. We are working in quadratic field ...
2
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0answers
320 views

Uses of Taylor series expansion [duplicate]

In calculus we use taylor series expansion at large number of places.I recently one of the application in number theory(To find solution of polynomial in finite field of order $p^{n}$, where p is ...
2
votes
2answers
196 views

Integer values of difference between cube and sqaure

I am trying to find square integer values for $k = a^3-b^2$ and $\gcd(a,b) = 1$ i.e. the values of and b for which k is a perfect square.
2
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0answers
243 views

Find all ideals with given norm

I'm working through a final exam from 2 years ago. First task was to find the ideal class group of $\Bbb{Q}(\sqrt{-73})$. That is not the difficult work. I can give the 4 representants of the group by ...
2
votes
1answer
137 views

Re-writing a series, involving prime numbers

Consider the limit $$\lim_{n \rightarrow \infty} \sum_{k=0}^n \frac{\Lambda(4k+3)}{(4k+3)}-\frac{1}{2}\ln(4n+3)$$ Where $\Lambda(n)$ is the vonmangoldt function, that is equal to zero if n is not a ...
2
votes
0answers
215 views

unique factorization of matrices

If I have a set of matrices, call this set U, how can I make this a UFD (unique factorization domain)? In other words, given any matrix $X \in U$, I would be able to factorize X as $X_1 X_2 ... X_n$ ...
2
votes
0answers
138 views

Integral of product of two square waves over [0,1]

In Mathematica notation, I am looking for the function f[m,n] for real numbers m and n defined by f[m_,n_]:=Integrate[SquareWave[m x]SquareWave[n x],{x,0,1}]. I'm trying to get a closed form for the ...
2
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0answers
62 views

Upper bound for non-square-free sum

Given a multiplicative function $f$, is there any general method of getting a upper bound of $\sum^{'}_{n<x} f(n)$, where the sum is restricted to all those non-square-free $n$? For example, when ...
2
votes
1answer
97 views

projectors in a tensor product of number fields

Let $F=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic number field and choose an ordering of the Galois group $Gal(F/\mathbb{Q})$, let us say $\{id, \sigma\}$. Then one has an isomorphism $F ...
2
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0answers
124 views

A natural way of thinking of the definition of an Artin $L$-function?

Emil Artin knew that given a finite extension of $L/\mathbb{Q}$, the local factor of the zeta function $\zeta_{L/\mathbb{Q}}$ at the prime $p$ should be $\displaystyle\prod_{\mathfrak{p}|p}\frac{1}{1 ...
2
votes
0answers
53 views

How many ways are there of coloring $n$ numbers? [duplicate]

Possible Duplicate: In how many ways can we colour $n$ baskets with $r$ colours? How many ways are there of coloring $n$ numbers $1, 2, 3, \dots, n$ ($n \ge 2$) in a circle $(C)$ with $p$ ...
2
votes
0answers
69 views

Some fast way to find a solution set to $us+vt \equiv 0 \pmod d$

1) Suppose that $u,v,d$ are nonzero, unequal fixed/given natural number. $s,t$ are nonzero natural numbers. We want to find a solution set $(s,t)$ that satisfies $us+vt \equiv 0 \pmod d$. What would ...
2
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0answers
59 views

Prove $\left(\frac{q(q+1)}{p}\right) =\left(\frac{1+q^{-1}}{p}\right )$ for $p\gt2$ a prime, and any $q \in \mathbb{Z^+} $.

For $p\gt2$ a prime, and any $q \in \mathbb{Z^+} $, Show that $\left(\frac{q(q+1)}{p}\right) =\left(\frac{1+q^{-1}}{p}\right )$ where the terms are legendre terms. I saw this result as part of a ...
2
votes
0answers
102 views

Searching for prime candidates

For some additional excitement, I've been searching for primes $p \gg q = 104729$, where $q$ is of course the ten-thousandth prime. It seems that the best way to search for prime candidates $p$ is to ...
2
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0answers
85 views

do decimal expansions of irrationals have applications?

I am not into number theory at all -- is there a specific reason why some researchers spend enormous effort on calculating millions of digits in the decimal (or any other base) expansions of ...
2
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0answers
79 views

A question on algorithm complexity

It is well-known that the evaluating the Discrete Fourier Transform definition directly has a complexity $O(N^{2})$ for a signal with bandwidth $N$. How to see or show that the fast Fourier transform ...
2
votes
1answer
151 views

Inclusion Exclusion and lcm

I would like to show that for any positive integers $d_1, \dots, d_r$ one has $$ \sum_{i=1}^r (-1)^{i+1}\biggl( \sum_{1\leq k_1 < \dots < k_i \leq r} \gcd(d_{k_1}, \dots , d_{k_i})\biggr) ~\leq~ ...
2
votes
0answers
354 views

Restriction on polynomial with integer coefficients

Let $P$ be a polynomial with integer coefficients such that for every positive integer $n$, $P(n)$ divides $2^n - 1$. Show that $P(x) =1$ or $P(x) = -1$ for all $x$.