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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1answer
54 views

factoring polynomials in $\Bbb Z/11\Bbb Z$

Any ideas as in how to Factor $x^{10}-1$ into linear factors in the integers modulo $11$, $\Bbb Z_{11}=\Bbb Z/11\Bbb Z$? I've been trying but can't come up with an answer.
3
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2answers
75 views

Finding the GCD of two Gaußian integers

How do you calculate the GCD of $6-17i$ and $18+i$ in $\Bbb Z [i]$?
2
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4answers
80 views

How did this method get the correct answer? Product of divisors

Find the product of the divisors of $420^4$. $420^4=2^8 \cdot 3^4 \cdot 5^4 \cdot 7^4$. I tried the following method which I soon realized was flawed: for $2^8$ there are $5^3$ possible divisors, for ...
1
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1answer
33 views

Euler product of Dirichlet series

Let $f$ be an arithmetic function such $f(n_1n_2)=f(n_1)f(n_2)$ for all $n_1,n_2 \in \mathbb{N}$ with $\gcd(n_1,n_2)=1$. Suppose we know that the Dirichlet series $$F(s) = \sum_{n=1}^{\infty}f ...
2
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2answers
14 views

How can I show that ord_n(a^u) is equal to t/(t,u)

I'm having a hard time on this question: Let n > 0, a $\neq$ 0, and (a,n) = 1. If ord$_n$ a = t, and if $u$ is a positive integer, then show that: $$ord_{n}(a^{u}) =\frac{t}{(t,u)}.$$ So far I've ...
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2answers
95 views

Integer solution to a square root equation

Given the following function: $$f(x) = \sqrt{36 x^2+16416 x +61084}$$ where x is an integer and the function also generates an integer value, is there an algorithm to determine its integer solutions?
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0answers
19 views

Complete residue mod $p$ and number of solution to an equation

Prove that there are infinitely many primes $p$ such that the total number of solutions $\pmod{p}$ to the equation $3x^{3}+4y^{4}+5z^{3}-y^{4}z \equiv 0$ is $p^2$. I can show that for $p \equiv ...
2
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2answers
87 views

Integer solutions of the equation: $x^2+y^2+z^2=kxyz$

Given the equation: $$x^2+y^2+z^2=kxyz$$ with: $(k,x,y,z)\in\mathbb{N}$, the only solution for $k=2$ is: $x=0,y=0,z=0$. For what values of $k$ the equations has solutions in which $x,y,z$ are ...
0
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1answer
49 views

Factorial Taxicab Number

What is the $i$th number $T_! (n,k,i)$ such that $T_! (n,k,i)$ is the sum of $n \in \mathbb{N}$ distinct positive integer factorials in $k \in \mathbb{N}$ distinct ways (ignoring ordering, ...
1
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3answers
67 views

Additive identity in $\mathbb{Z}_{7}$

The problem consists of two parts: 1)Prove that for some integer $m,$ $3^{m} = 1$ in the integers modulo 7, $\mathbb{Z}_{7}$, which i have done. The second part asks to use this fact to deduce that ...
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2answers
259 views

Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)

Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer. I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
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0answers
31 views

Rank four quadratic form with trivial discriminant

Is there an example of a field $k$, quadratic form $\varphi$ of rank four, which is anisotropic over $k$, has trivial discriminant and is not a Pfister form? In case of rank six one can use Albert ...
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2answers
86 views

number theory proof regarding number of roots

I was given this proposition but I was never able to prove it. Does anyone know how to solve this? if f is a polynomial in $\Bbb Z_p\left[x\right]$ and the deg(f) = n then f can have at most n roots. ...
2
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1answer
53 views

Understanding a proof of Lagrange's four-square theorem

I've been looking at Wikipedia's proof of the four-square theorem and trying to work out the details - I like that it doesn't need to separate the cases for $m$ even and odd, but there is one step ...
5
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2answers
116 views

Is this Goldbach-type problem easy to solve?

Problem: Given an odd prime number $p$, are there odd prime numbers $q$, $p'$, $q'$ such that $\{p,q\} \neq \{ p',q'\}$ and $p+q = p'+q'$ ? This comment informs that it's an obvious ...
0
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1answer
62 views

number theory proofs with units, orders, and the phi function

How do you prove the following? : There is an element $u_0$ of $U_m$ whose order is divisible by the order of every other element of $U_m$. If the order of $u_0$ is n then the polynomial $x^{n-1}$ ...
-1
votes
1answer
178 views

Number of ways to win chocolate game

Alice and Bob are playing a game. They have N containers each having one or more chocolates. Containers are numbered from 1 to N, where ith container has A[i] number of chocolates. The game goes like ...
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2answers
58 views

The existence of certain numbers

i will try to make my question clear. Whe know the existence of $\phi=1.61803398874989484_\ldots$ or $\pi=3.14159265359_\ldots$ and we know that the decimals numbers goes to infinity. But, let's say ...
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0answers
36 views

The middle coefficient of a cyclotomic polynomial

It is known that the middle coefficient of a cyclotomic polynomial $\Phi _n$, with $n\geq 3$, is $0$ (when $n$ is a power of 2) or an odd integer. Is it known just what odd integers can occur?
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1answer
25 views

Regarding Thue's congruence theorem.

Did the mathematician Thue have a theorem where if $X\cdot N$ is congruent to $y \pmod m$, gcd$(y'm)=1$ then $1 \lt X \le \sqrt{m}$, or $1 \lt Y \le \sqrt{m}$? I'm not sure if I saw this in a number ...
2
votes
3answers
92 views

Trouble with inequalities

I'm a 9th grade student, going into 10th grade. Math has always been a subject I enjoyed and excelled in. I'm writing a schoolboard-wide math contest next year in mid-February I believe. To prepare ...
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3answers
68 views

Number theory proofs relating to divisors [closed]

How do you prove this? $$\left(n-1\right)^2\mid\left(n^k-1\right)\Longleftrightarrow\left(n-1\right)\mid k$$
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3answers
109 views

Using decimals of $\pi$ to store data

I read recently about an idea to, instead of storing actual data, converting the data to a string of digits and then store the index of where this pattern occurs in some number, for example $\pi$. The ...
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0answers
46 views

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ primes. What are the first values of $U(n)$?

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ prime numbers (except for the first prime number: $2$). What are the first values of $U(n)$ up to ...
1
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1answer
55 views

Number theory proofs relating to GCD's [duplicate]

$\left(a^n-1,a^m-1\right)=a^{\left(n,m\right)}-1$ for positive integers a,n,m where (a,b) stands for the GCD of a and b How do you prove this?
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1answer
46 views

Bound on $a,b,c,d$ and truth of statement $\forall \, x \in \mathbb N, \exists \, a,b,c,d \le {l(x)} \in \mathbb N:x=a^b+c^d$

To render the statement in the title in relatively simple English, "For every natural number $x$, there exist four natural numbers $a,b,c,d \le l(x)$, where $l(x)$ is a function of $x$ which bounds ...
0
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2answers
70 views

if each each of two natural numbers a and b is a sum of two squares then ab is also a sum of two squares

How do you prove if each each of two natural numbers a and b is a sum of two squares then ab is also a sum of two squares I'm pretty sure that this is a true statement but I don't know how to prove ...
3
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1answer
50 views

Identities for L-series and Euler product

It is a mabe a stupid question for many experts here. There is something wrong in the following reasoning, and now I could not find it. Could someone help me out? Any advice will be highly ...
5
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1answer
336 views

Identity for frequency of integers with smallest prime(n) divisor

An identity for A038110 and A038111: $$ \frac{\phi(e^{\psi(p_{n}-1)})}{e^{\psi(p_{n})}}=\frac{\prod _k^{n-1} \left(1-\frac{1}{p_k}\right)}{p_n}, $$ where $\psi(\cdot)$ is the second Chebyshev function ...
0
votes
1answer
26 views

Linear Constraints Solution Existence

how can one decide if $$A*t\ge b$$ $A$ is a Matrix with integer Entries and $t$ is a Vector with integer Entries, $b$ is a fixed Vector with integer Entries exists?
6
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3answers
116 views

The relation between the number of $0$s which are at the end of $3^{n!}-1$ and that of $n!$

Let $a_n,b_n$ be the number of $0$s which are at the end of $3^{n!}-1,n!$ in the decimal system respectively. I found that $a_n=b_n+1$ holds for $n=4,5,\cdots, 10$. Then, my questions are... ...
3
votes
1answer
63 views

A problem from Komal

For every integer $n\ge 2$ let $$P(n)=\prod (\pm \sqrt{1} \pm \sqrt{2} \cdots \pm \sqrt{n})$$ where the product is over all possible permutations of the signs. Prove $P(n)\in \mathbb{Z}\;\forall ...
0
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2answers
27 views

Divisors of the product of two coprime integers can be written as the product of two coprimes

In my lecture notes: Let $m,n\in \mathbb{N}$ be relatively prime. The fundamental theorem of arithmetic implies that each divisor of $mn$ is the product of two unique positive relatively prime ...
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2answers
53 views

An exercise in number theory: euclidean domain

I have an exercise for you about euclidean domain. Which primes $p<30$ in $\mathbb{Z}$ is a prime in $ \mathbb{Z} \left[ \frac{1+\sqrt{-7}}{2} \right] $ ? Thank you very much for the support, I ...
0
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4answers
54 views

Phenomenon regarding square of any integer…

There is a phenomenon regarding squares of integers which i observed today. $n^2 = \sum_1^n^-^1 + \sum_1^n $ I am a computer science graduate and i never heard about this phenomenon till date. Is it ...
5
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0answers
37 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 ...
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0answers
88 views

Number theory proofs relating to units

Moderator Note: This has been claimed to be a current contest question. It is being locked while we investigate. What is a counterexample for the proposition: If u ∈ Um has order n1 and u2 ∈ Um ...
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2answers
122 views

How to find the roots of polynomials in $\Bbb Z_p$

Let $p$ be a positive prime. What are the roots of the polynomial $x^{p-1} - 1$ in $\Bbb Z_p$? Factor this polynomial into linear factors in $\Bbb Z_p[x]$. What does this factorization tell you about ...
2
votes
1answer
132 views

A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

I have found a closed form for the following new series involving non-linear harmonic numbers. Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = ...
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2answers
40 views

Proofs regarding units in number theory

I was given this problem to prove and I have no idea how to prove it: How do you prove that if $u \in U_m$ has order $n$ and if $n=kg$, than $u^k$ has order $g$. Any ideas?
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0answers
24 views

On certain error terms

Define $$E_k(x)=-\sum_{d|k}\mu(d)B_1\left(\left\{\frac{x}{d}\right\}\right)$$ where $\mu(d)$ is the mobius function, and $B_1(x)$ is $x-1/2$ and $\{x\}$ is the fractional part of $x$. define $s(k)$ ...
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0answers
19 views

Integer Solution To System of Linear Function

Hello i reposted the question, because the equation title was a bit misleading. [and i could not edit the question] $c,f$ are not known, so it is not a Diophantine equation, it is more like a system ...
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0answers
23 views

products of bernoulli poynomial

prove the following assuming that $d|k$, $e|k$, $(d,e)=gcd(d,e)$, $B_1(x)=x-1/2$, $\{x\}$ is the fractional part of $x$ $$ \int_0^k ...
3
votes
1answer
47 views

What are the conditions needed by $n$ so that $2^{n+1}5^n-1$ is a prime number?

What are the conditions needed by $n$ so that $2^{n+1}5^n-1$ is a prime number? I am investigating prime numbers $2^{n+1}5^n-1$ for some $n\in\mathbb{N}$ and I found out that there are prime numbers ...
2
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0answers
45 views

Integer Solutions To Linear Equation

$$a*q_1+b*q_2=c$$ $$a*q_3+b*q_4=f$$ $q_1, q_2, q_3, q_4$ rational numbers, $c,f$ integer Given $q_1, q_2$ can you construct all solutions $(a,b)$ where $c,f$ is intenger I made an edit since the ...
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4answers
72 views

Finding roots of equation in $Z_{14}$, $Z_{17}$

Find all the roots of the equation $x^2 - 9x + 6 = 0$ in $Z_{14}$. Factor the polynomial $x^2 - 9x + 6$ into linear factors in $Z_{14}[x]$ in all possible ways. Find all roots of $x^2 - 9x + 3 = 0$ ...
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2answers
58 views

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3…10\}$?

How to find a polynomial with $f(1), f(4),f(9)$ prime and coefficients in $\{1,2,3...10\}$? I can't think of any way on how to generate such types of polynomials? Also, would they have a minimum ...
0
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2answers
33 views

Finding Solutions to a Diophantine Equation with Factorials

How many ordered pairs of positive integers $(a, b)$ are there such that $a!+\dfrac{b!}{a!}$ is a perfect square? Is the number of solutions finite? Source: Ran into it on Facebook. I have plugged ...
29
votes
1answer
443 views

Very tight prime bounds

Is it possible that $$\left|\operatorname{li}(n)-\sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}-\log(2)-\dfrac{1}{2}\right|<\dfrac{2\sqrt{n}}{e\log(n)}?$$ Since $$ ...
3
votes
1answer
36 views

Showing two numbers are relatively prime in number fields

Solve $x^3-2=y^2$ in integers. The standard way to solve this problem is to consider the arithmetic of the ring of algebraic integers $\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}$ and to show that ...