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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Enquiry on the Riemann $\xi$ function.

The Riemann xi function, $\xi(s)$, is known to be real valued on the critical line $s=1/2 + it$ where $t$ is real. But is it also real valued when $t$ is complex, that is, of the form $a+bi$ for some ...
6
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0answers
26 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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28 views

Prove that the set contains $\Phi$ elements

Let $x \in \{0,1,\ldots,n-1\}$ where $n$ is a positive squarefree number. If $x$ is relatively prime to $m$ and $\Phi$ is the number of such $x$, prove that the set $\{a | x^m \equiv a \pmod{n} ...
0
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0answers
28 views

Questions that SAGE, MAGMA can answer?

I practice theoretical mathematics and I know (almost) nothing about SAGE, MAGMA. I would like to know (in general) what type of questions can I ask SAGE to do? For example, I know that given an ...
3
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2answers
40 views

Does $p=x^2+4y^2$ imply that $x$ is a quadratic residue mod $p$?

Does $p=x^2+4y^2$ imply that $x$ is a quadratic residue mod $p$? I'm stuck on this problem. My attempt: We know that since $p$ is a sum of squares $p\equiv 1 (4)$. This means that ...
1
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1answer
25 views

Finding the smallest composition of a natural number with limited basic set of summands

W.l.o.g. I have a set of natural numbers $$S = \{s_1, \ldots, s_n\}, \quad s_i \in \mathbb N$$ as well as an $x \in \mathbb N$ I would like to express as sum of $s_i$. How do I find the smallest ...
0
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0answers
8 views

Are these value groups equal?

Assume that a field extension $L/K$ is finite, and K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ ...
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2answers
30 views

axiom of continuity guarantees no gaps exist on the real axis?

I read this content at the bottom of this page just wonder why axiom of continuity could guarantee no gaps exist on the real axis? any proofs ?
4
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0answers
40 views

Show that any positive rational number can be expressed as $\frac{a^3+b^3}{c^3+d^3}$. [duplicate]

Show that any positive rational number can be expressed as $$\frac{a^3+b^3}{c^3+d^3}$$ Perhaps the statement means that for every two positive integers $m$ and $n$, there exists a $k$ such that ...
2
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2answers
123 views

Do we know the value of $3 \uparrow\uparrow\uparrow 3$

I was studying Graham's number and before we can even start calculating G1 which is $3\uparrow\uparrow\uparrow\uparrow 3$, I was wondering if we even have the actual value of $3 ...
0
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1answer
10 views

How to permute remainders of CRT between residue classes?

I want to know how can be permuted the remainders of the CRT. How to go from $a \equiv r1 \;(\bmod\; n_1)$ $a \equiv r2 \;(\bmod\; n_2)$ to $b \equiv r1 \;(\bmod\; n_2)$ $b \equiv r2 ...
4
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2answers
60 views

Interesting and unusual word problem with prime numbers and factors

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with prime numbers, but other than that, the textbook gave no hints really and ...
0
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1answer
19 views

Entry of $1-9$ in magic box

There are 9 slots to fill. Question ask us to fill it using $1-9$ each being used only once. But what I can see here is that $5th$ column must be filled with $1,2$ and $3$ but after $1,2$ and $3$ ...
0
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2answers
51 views

Using remainder theorem in Pythagoras theorem makes absurd results!

At first, I apologize for the title. I really couldn't find anything better than this. Now,we know, some integers $a,b,c$ (none of them are $0$) can be found so that $$a^2 = b^2 + c^2$$ Now,here,of ...
1
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1answer
31 views

When does a binomial have repeated roots mod p?

Given a polynomial $f(x)=x^n+a$, and I have that $p$ does not divide $an$, can I show that $f(x)\pmod p$ has no repeated roots? I'm not sure how to proceed.
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0answers
51 views

Lower bound related to Goldbach conjecture

I am curious to know if a lower bound on the number of ways (call this $\beta$ and assume $p_1 + p_2$ distinct from $p_2 + p_1$) in which two primes $p_1, p_2$ that add up to a given even integer $n$, ...
0
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1answer
24 views

Power Tower modulo sequence

A power tower, defined here is $a\uparrow\uparrow n$. ($a\uparrow\uparrow 2=a^a$, $a\uparrow\uparrow 3=a^{a^a}$, $a\uparrow\uparrow 2=a^{a^{a^a}}$, etc...) Is there a base $a$ such that ...
0
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0answers
48 views

A Product of Primes [on hold]

Anyone ever seen a function like this: F(1)=1st prime, F(2)=2nd prime * 1st prime, F(n)=nth prime * F(n-1)?
4
votes
3answers
179 views

Geometric Interpretation of the Basel Problem?

I'm probably asking a question no one knows the answer to, but everyone must mentally ask at some point. Does the Pi in the solution to the Basel problem have any geometric significance? Every time ...
0
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1answer
38 views

On a certain prime structure.

It is unknown whether there are infinite primes $p$ where $2p-1$ is also a prime. Is it known there are only finitely many primes $p$ such that both $q$ and $2p-1$ are primes where $p-1=2aq$ for any ...
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0answers
25 views

Prove that $0,1,b_1,\ldots,b_{n-2}$ are all distinct numbers

Let $n$ be a positive squarefree number. Prove that $0,1,b_1,\ldots,b_{n-2}$ are all distinct numbers where \begin{align*}0^m &\equiv 0 \pmod{n} \\1^m &\equiv 1 \pmod{n} \\2^m &\equiv ...
7
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1answer
77 views

Repeatedly taking mean values of non-empty subsets of a set: $2,\,3,\,5,\,15,\,875,\,…$

Consider the following iterative process. We start with a 2-element set $S_0=\{0,1\}$. At $n^{\text{th}}$ step $(n\ge1)$ we take all non-empty subsets of $S_{n-1}$, then for each subset compute the ...
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0answers
21 views

How can we find all solutions to a Pell-type equation?

Is it true that for solveable Pell-type equations, all solutions are given by: 1: Finding fundamental solution to Pell's equation 2: Find all solutions of the Pell-type equation less then the ...
1
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1answer
86 views

continued fraction $F(x)$ that is a generating function of central binomial coefficients

Given the following continued fraction $$F(x) =\cfrac{1}{x+\cfrac{2^2(2^2-1)}{6x+\cfrac{3^2(3^2-1)}{12x+\cfrac{4^2(4^2-1)}{20x+\cfrac{5^2(5^2-1)}{30x+\ddots}}}}}=\frac{1}{\sqrt{x^2+4}}$$ Then ...
2
votes
1answer
40 views

zeroes of homogeneous analytic $p$-adic functions

I am trying to understand Lemme 2.1 page 3 of this paper by Pilloni. What is says (I think) is that if you have, for a a positive real number $w$, an analytic function $$ f : ...
2
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1answer
76 views

Find all solutions to the Diophantine equation $x^2-7y^2=-3$

I want to find all integer solutions of the equation $$x^2-7y^2=-3$$ I don't really know where to start... I tried the one trick I know which is to factor in some quadratic ring: ...
2
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0answers
19 views

Iterate Over Integer Partition Refinement in Sage

A partition of an integer $n$ is a non-decreasing list of positive integers summing to $n$. For example, $3$ can be partitioned as $1 + 1 + 1$, $1 + 2$ or just $3$, but $2 + 1$ is indistinct from $1 + ...
7
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0answers
170 views

Seemingly easy Diophantine equation $a^3+a+1=3^b$

How to prove that $a=b=1$ is the only positive integer solution to the following Diophantine equation?$$a^3+a+1=3^b$$
0
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0answers
12 views

Integral formula for the local L factor of a base changed automorphic representation

Let $\Bbb A$ the ring of rational adeles and let $\pi=\bigotimes_{p\leq\infty}\pi_p$ be an automorphic (cuspidal) representation of ${\rm GL}_2(\Bbb A)$. Fix a quadratic extension $K\supset\Bbb Q$. ...
2
votes
3answers
43 views

Finding all natural solutions to $2^a+5=b^2$

What are all possible natural number solutions $(a, b)$ to the equation $$2^a+5=b^2 ?$$ The only solution I've found is $$2^2+5=3^2.$$ This may be the only solution but I don't have a proof.
3
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1answer
199 views

conjectured general continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following general continued fraction which is symmetric on $a$ and $b$ is ...
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0answers
26 views

The largest product of two n-digit numbers which is palindrome

Project Euler: 4 is stated as follows: Largest palindrome product A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 ...
2
votes
2answers
35 views

Difference of subsets of integers with $A-A=2 \mathbb{Z}\setminus \{-2k,2k\}$

Is there any subset $A$ of integers such that $A-A= 2\mathbb{Z}\setminus \{-2k,2k\}$, for some integer $k$? ($A-A=\{a_1-a_2: a_1,a_2\in A\}$, and $2\mathbb{Z}$ is the set of even integers.)
5
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2answers
77 views

Can we describe nicely all the rational numbers of the form $x^{2}-xy+y^{2}$?

Let $\zeta$ be a primitive cubic root of unity, i.e., $\zeta$ is a complex number such that $\zeta^{3}=1$ and $\zeta\neq1$. Let us consider the Galois extension $\mathbb{Q}(\zeta)/\mathbb{Q}$ (the ...
1
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1answer
75 views

Given $x_m=4x_{m-1}-x_{m-2},\ x_1=1,\ x_2=3$

I friend told me that apart from trivial ones, the elements in this sequence never equal powers of 3: $$x_m=4x_{m-1}-x_{m-2},\ x_1=1,\ x_2=3.$$ Could you please help me to prove this?
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44 views

A question about square roots of quadratic residues.

Suppose $\mathbb{Z}_p^*$ ($p$ is a prime) is a cyclic group with generator $g$. We consider a subgroup $\mathbb{G}$ of $\mathbb{Z}_p^*$ with generator $h$ and order $q$, where $h = g^4~mod~p$ and ...
0
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0answers
16 views

Finding the number of integer points inside a sphere of radius R and dimension D centered at the Origin

I am writing a computer program to count the number of integer points inside a sphere of radius R and Dimension D centered at the origin. In essence, if we have a sphere of dimension 2 (circle) and ...
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0answers
32 views

Power of 3 with all odd digits?

Is the largest power of 3 with all odd digits 9? I can't seem to find a power of 3 that contain all odd digits except 3 and 9. (I checked the first 200 powers, and there is at least one even digit.) A ...
2
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1answer
100 views

Find all $x,y,z \in \mathbb{Z^{+}}$ such that $20^x+15^y=2015^z$

Find all $x,y,z \in \mathbb{Z^{+}}$ such that $20^x+15^y=2015^z$ I was checking modulo $4$ to see that $y,z$ must have the same parity. Then took two cases when both $y,z$ are even and when both ...
0
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0answers
13 views

for a chen prime p, what is the size of factors of p+2

Suppose the twin prime conjecture fails. Then, by Chen's theorem, there are infinitely many primes $p$ s. t. $p+2$ is a product of exactly two primes. It would be nice to know that as $p$ grows, so ...
0
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1answer
19 views

Proving that if gcd(m, n) = 1, and if d divides mn, then there exist unique numbers a and b such that a divides m, b divides n, and d = ab.

What do I know? If d | mn, there exist an integer k such that dk = mn. I also know that because gcd(m, n) = 1 there exist some integers x and y such that mx + ny = 1. I am having trouble to prove ...
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0answers
31 views

Prove that the numbers are different

Let $n$ be an integer that is not divisible by any square greater than $1$. Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n$. If a period $t$ of $x_m$ is ...
0
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0answers
10 views

To find out HCF of two numbers [duplicate]

Given that $(a,b)=1\ \text{and}\ p\ \text{is odd prime,then prove that }(a+b,\frac{a^p+b^p}{a+b})\ \text {is 1 or}\ p$. have no idea where to start.any hint please
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1answer
1k views

How to determine the key-matrix of a Hill cipher where the encrypted-message-matrix is not invertible?

I am new to this subject and I have a homework problem based on Hill cipher, where encryption is done on di-graphs (a pair of alphabets and not on individuals). The alphabet domain is $\{A\dots ...
3
votes
1answer
12 views

Separable polynomial with splitting field an unramified extension?

I am trying to prove a theorem and it seems that I need that an irreducible polynomial $f(x)$ that is separable over $\mathfrak{p}$ has its splitting field an unramified extension of ...
0
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0answers
17 views

Solutions in $\mathbb Q_p$ leads to solution for congruences equations?

Let $p$ be a prime number such that $p\equiv1\pmod 3$. Let $n$ be an integer such that the equation $x^3=n$ has a solution in $\mathbb Q_p$. In fact with our assumptions, the others solution are in ...
3
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0answers
35 views

Pairwise distinct subsequence

Suppose there is an infinite sequence $S_n = (s_1, s_2, \dots )$ generated by a finite set of numbers $\{1, 2, \dots, n\}$. Given a number $m$ such that $m < n$, the subsequence $(s_i, s_{i+1}, ...
1
vote
3answers
68 views

Number theory contradiction

I was solving this question How many pairs of integers $(a,b)$ that satisfies the following conditions: $1\le a,b\le 42$ $a^9\equiv b^7\pmod{43}$ and I ran into a contradiction: We ...
3
votes
1answer
31 views

Determining when ring of integers is $\mathbb{Z}[\theta]$

Something which is not difficult to prove is that if $K$ is a number field generated by an integer $\theta$, then the ring of integers $\mathfrak{O}_K$ is generated over $\mathbb{Z}$ by $\theta$ and ...
0
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1answer
25 views

About $\sum_{i\geq 1}\frac{1}{(n+i)_{n+1}}$ and $\sum_{i\geq 1}\frac{1}{i^2-i-1}$

I was playing around with Zeta function and changed it as following to find that $$\sum_{i=1}^{\infty} \frac{1}{i\cdot(i+1)\cdot(i+2)\cdot\ldots\cdot(i+n)} = \frac{1}{n\cdot n!}$$ ...