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Questions tagged [number-theory]

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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10 views

Prove $F^* = \mu * F$

Let $f: \mathbb{Q} \cap [0,1] \to K$ and set $F(n) = \sum_{k = 1}^n f(\frac k n)$, $F^*(n) = \sum_{k = 1, (k,n) = 1}^n f(\frac k n)$. Show that $F^* = \mu * F$ where $*$ is the Dirichlet product....
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2answers
61 views

Solve $a^3+b^3+3ab=1$ with $(a,b)\in \Bbb{Z}^2$

Solve the following equation for $(a,b)\in \Bbb{Z}^2$: $$a^3+b^3+3ab=1$$ I tried all of the standard techniques I know. I tried modular arithmetic: $$a^3+b^3+3ab\equiv 1 \pmod{3} $$ $$a^3+b^3\...
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2answers
42 views

Questions on Dirichlet's approximation theorem

In Wikipedia, the entry Dirichlet's approximation theorem states as follows: In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, ...
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2answers
41 views

How many triplets of consecutive primes $p$, $q$, $r$ are such that $p^2+q^2+r^2$ is also prime?

So we have $p$, $q$, $r$ which are successive prime numbers, we wonder how many such triplets are such that $p^2+q^2+r^2$ is also a prime. My attempt: I tried to $p^2+q^2+r^2$ refactor it somehow, ...
0
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1answer
43 views

Sum of five-digit number is 10 problem

So sum of five-digit number is 10. And all the digits are different from each other. If we sum this number with a number written in reversed order of those digits, we will get a number that has same ...
0
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0answers
18 views

Is there a functor which is equivalent to discriminant of number field?

Let $K$ be a number field, i.e. a finite extension of $\mathbb{Q}$. The ring of integer $O_K$ is a free $\mathbb{Z}$-module. Let $\{ a_1, \cdots , a_n\}$ be a integral basis of $O_K$. Then, $$ \...
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0answers
32 views

Number of partition of $n$ into sum of three squares, needing fast algorithm

Does anyone know how to find number of partitions of $n$ into sum of three squares with asymtotic time complexity faster than $O(n)$ ? I found some info on https://oeis.org/A000164, but there are an $...
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0answers
18 views

Number of ordered pairs dividing 720

How many ordered pairs of natural numbers like $(a,b)$ exists such that $a$ and $b$ divide 720 but $ab$ doesn't? my own idea was to solve in two cases: 1.a==b and 2.a!=b I found answer 24 pairs for ...
2
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1answer
54 views

numbers that cannot be expressed in closed form?

Irrational numbers can be divided into two categories: - Algebraic - Transcendental But there are some numbers that are roots of polynomial ie. are algebraic but cannot be expressed in closed ...
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1answer
37 views

Non-negative integer solution for $ax + by = c$ [on hold]

Show that if positive integers $a$ and $b$ are relatively prime, then every integer $c > ab$ has the form $ax + by = c$, where $x$ and $y$ are non-negative integers. According to the common way of ...
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3answers
53 views

Describe all integral solutions of the equation $x^2 + y^2 = 2z^2$ such that $x,y,z > 0$, gcd$(x,y,z) = 1$, and $x > y$.

As the title states, the question tasks me with finding all the integral solutions of the equation under the specified constraints. I have an idea of where to start due to a somewhat similar problem ...
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1answer
34 views

proving the fundamental theorem of arithmetic

The book is trying to prove the fundamental theorem of arithmetic but does he mean by saying that at least one of the inequality must be true with strict inequality ? And from where did he come up ...
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2answers
41 views

Partitioning $[1,v-1]$ into sets of size three such that the sum of each set is $3v/2$

Suppose that $v \equiv 4\;(\bmod 12)$. In general, is it possible to partition the integer interval $[1,v-1]$ into integer partitions of $3v/2$ with three distinct parts (with each part in $[1,v-1]$)? ...
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0answers
43 views

Why is the subject of math overall considered an essential for compulsory education/non-scientific circles/society overall? [on hold]

It would seem that math is pushed both scientifically and culturally as some sort of plethora of necessity. What really does math do for the layperson? Aside from basic counting and stuff, I can't ...
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46 views

Does $a^a+b^b=c^c$ have any solution in $\mathbb{N}$? [on hold]

Just as the question implies. i'm curious if this equation have a solution contrary to Fermat Last theorem.
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1answer
70 views

Does digit $6$ always lead to $\ 25921=161^2\ $?

Consider prime numbers with the property that the product of the factorials of the digits plus $1$ is a perfect square, for example the prime $$30241$$ leads to the square $$3!\cdot 0!\cdot 2!\cdot 4!\...
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0answers
27 views

Clarification on this continued fractions question.

I would rather this not be solved for me , as its a homework question , I just want some clarification on whether I'm understanding it correctly. Say we have the continued fraction $\alpha=[3,\bar{2}...
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0answers
22 views

Find the numbers of ordered array $(a,b,c,d)$ such $a^2+b^2\equiv c^3+d^3\pmod p$

Let $p$ be prime number,and such $p\equiv 1\pmod {12}$,Find the numbers of ordered array $(a,b,c,d)$ that satisfies the following conditions: (1):$a,b,c,d\in \{0,1,2,\cdots,p-1\}$ (2):$a^2+b^2\equiv ...
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0answers
29 views

Question about A Formula for factoring Fermat Numbers

Given Fermat Number $$N=2^{2^{k-1}}+1$$ I have a formula such that inputting N and k into this formula it either: (A) Gives a proper factorization N=e*s when N is composite, and equals N when N is ...
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1answer
42 views

Easy proof of falsehood of $\pi(n) \leq C \cdot \text{ln}(n)$ for the prime counting function $\pi$

Let $\pi(n)$ be the number of primes in the range $1,\dotsc,n$. The following statement is true: There is no $C>0$ such that $\pi(n) \leq C \cdot \text{ln}(n)$ for all $n\geq 1$. It follows ...
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0answers
27 views

Ramification Examples

Any help for seeking a totally tamely ramified extension of $\mathbb{F}_3$((X)), an unramified extension of $\mathbb{F}_3$((X)), a totally tamely ramified extension of $\mathbb{Q}_3$(i)? Feel ...
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1answer
20 views

Understand lattice definition and volume , algebraic number theory

I am taking the course algebraic number theory ( book J. Neukirch algebraic number theory ) and we reached a point where we are talking about the lattice I know that a lattice is a discrete subgroup ...
0
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1answer
42 views

Finding the quadratic polynomial of a continued fraction

Given $\alpha=[3,\bar{2},\bar{4},\bar{5}]=3+\tfrac{1}{2+\tfrac{1}{4+\tfrac{1}{5+...}}}$ I want to (i) find the quadratic integer polynomial P for which $\alpha$ is a root and (ii) hence write $\alpha$...
4
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3answers
119 views

Why is this proof of a congruence relation valid?

The following question comes from the 2012 Singapore Mathematical Olympiad (Open Section), Round 2. Let $p$ be an odd prime. Show that $$1^{p-2}+2^{p-2}+3^{p-2}+\dots+\left(\frac{p-1}2\right)^{p-2}=...
1
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1answer
28 views

Solving a congruence mod 255 [on hold]

I would like to find the integer solutions of $128x^2+y^2 \equiv 0 \mod 255$
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0answers
22 views

question of legendre symbol [on hold]

Show that $(1|p) + (2|p) + (3|p) + \cdots + (p − 1|p) = 0$, where $p$ is odd prime. (Hint: Consider the total number of quadratic residues modulo $p$)
4
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3answers
107 views

prime numbers and expressing non-prime numbers

My textbook says if $b$ is a non-prime number then it can be expressed as a product of prime numbers. But if $1$ isn't prime how it can be expressed as a product of prime numbers?
2
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0answers
28 views

Looking for scientific papers about lotteries

Do you know any paper focusing on the statistical science and math behind lottery and number guessing games? I am creating a new one, where the winning numbers are known in advance, and the gain is ...
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4answers
88 views

If $ n^2+2 $ and $ n^2-2 $ are prime numbers, then $n$ is divisible by $3$.

If $n>1$ ($n \in \Bbb N$) and,if $ n^2+2 $ and $ n^2-2 $ are prime numbers, prove that $n$ is divisible by $3$. I have been trying to solve this with no success.
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1answer
14 views

Can a variant of the Dirichlet eta function converge for negative numbers?

It is known that the Dirichelt eta function, defined by $$ \eta(s)=\sum\limits_{n=1}^\infty\frac{(-1)^{n+1}}{n^s} $$ converges conditionally on the open half-plane $\Re(s)>0$. This fact inspires ...
3
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3answers
112 views

Is there any rational number $x \ne 0$, where $e^x \in \Bbb{Z} $? [on hold]

Is there any rational number $x \ne 0$, where $e^x \in \Bbb{Z} $ ? How would we prove this? Though this is quite an elementary seeming question I haven't been able to find any proof for it.
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1answer
50 views

The limit $\displaystyle \lim_{s\to 1}\left(\zeta_K(s)-\dfrac{\pi}{4(s-1)}\right)$.

Let $K=\mathbb{Q}(i)$ be the field of Gaussian rationals. Let $\zeta_K(s)$ be the Dedekind zeta function associated to K, defined by $\displaystyle \zeta_K(s):=\sum_{\mathfrak{a}\subseteq \mathbb{Z}[...
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0answers
26 views

Citizens and rebels: a twin prime related categorization of composites

Assuming Goldbach's conjecture and denoting by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ for a large enough composite integer $n$, consider the sequence $(u_n)_n$ such that $...
2
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2answers
58 views

What is the probability that $\exists N \in \mathbb{N}$ such that $\forall n > N$, $2n \in C + C$?

Suppose $C$ is a random subset of $\mathbb{N}\setminus\{1, 2\}$, such that $\forall n \in \mathbb{N}\setminus\{1, 2\}$, $P(n \in C) = \frac{1}{\ln(n)}$ and the events of different numbers belonging to ...
1
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1answer
17 views

Bound of logarithmic integral function

Does logarithmic integral function bound from above the prime-counting function? In other words, does it hold that $\pi(x) \le li(x)$ or even $\pi(x)\le Li(x)$?
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0answers
36 views

the isomorphism of $\mathbb{Z}_{p}[x] /\left(x^{n}-1\right)$ using Hensel's Lemma

I an trying to prove the following.Let p:prime, $n\in\mathbb{N}$ with $(n,p)=1$ ,and $x^{n}-1=f_{1}(x) \cdot \ldots \cdot f_{r}(x) \quad\left(f_{1}(x), \dots, f_{r}(x) \in \mathbb{Z}_{p}[x]\right.$ ...
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2answers
57 views

Prove $g$ is a generator if $g^q=1 \pmod p$

Let $p$ be an odd prime and let $q=\frac{p-1}{2}$. If $g^q=-1\pmod p$ and $q$ is a prime, show that $g$ is a primitive root mod $p$. I want to show $\{g,g^1,...,g^{p-1}\}=\{1,2,...p-1\}$. I argue by ...
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0answers
33 views

Grothendieck Trace Formula Reference [on hold]

Does anyone have a good reference that would fully explain everything that is going on in the Grothendieck Trace formula (https://en.wikipedia.org/wiki/Grothendieck_trace_formula) assuming little to ...
6
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3answers
218 views

Will these geometric means always converge to $1/e$?

Let $p_n$ be the $n$-th prime and $F_n$ be the $n$-th Fibonacci number. We have $$ \lim_{n \to \infty}\frac{(p_1 p_2 \ldots p_n)^{1/n}}{p_n} = \lim_{n \to \infty}\frac{\{\log(F_3)\log(F_4)\ldots \...
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2answers
84 views

How would one explain the concept of a p adic number in layman's terms?

The concept of a p adic number is something that I haven't quite been able to grasp. Heretofore, I can comprehend the fact that the p adic number system essentially extends the conventional arithmetic ...
2
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0answers
26 views

Bounds for quotient of two amicable numbers

Let $a,b$ be amicable with $a$ being the smaller of the two numbers. Are there any bounds for $\frac{a}{b}$ ? I looked online and found that there were maximal and minimal bounds but these do not ...
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3answers
41 views

If a,b are amicable numbers show that $(\sum_{d|a} d^{-1})^{-1} + (\sum_{d|b} d^{-1})^{-1}=1$

Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive integer divisor other than ...
6
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2answers
101 views

Show that $(\binom{p^2}{p} -p ) $ is divisible by $p^5$, for every prime number $p, p\ge 5$

Show that $(\binom{p^2}{p} -p ) $ is divisible by $p^5$, for every prime number $p, p\ge 5$. I have a combinatorics problem, and this is what it reduces to. I am not quite sure how to link the fifth ...
2
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1answer
24 views

Sums of divisors of an odd perfect square

Let $n^{2}$ be an odd perfect square. Under what conditions will: $\sigma (n^{2})\equiv 0 \mod 5$ where $\sigma $ is the sum of divisors function.
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0answers
31 views

Is the Euler-Mascheroni constant irrational? [on hold]

Please prove or disprove succinctly. $\gamma=\lim_{n\to\infty}[{H_n-\ln{n}}]$
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0answers
40 views

How could factordb apply the p-1-method on this number?

The following partial factorization http://factordb.com/index.php?id=1100000001285565404 was used by factordb to prove this number ($\ 32^{2133}+4^{2133}+1\ $) to be prime. http://factordb.com/...
4
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1answer
38 views

Topological Algebraic Independence of power series

Let $p$ be a prime number, let $x$ be a variable, and consider two power series over the ring $\mathbb{Z}_p$ of $p$-adic integers: $a(x):=\underset{n\geq 1}{\sum}{\frac{p^n}{n!}x^n}=px+\frac{p^2}{2}x^...
2
votes
1answer
45 views

Congruence and quadratic residues

Let $p_1$, $p_2$, $p_3$ be distinct primes satisfying $p_1 \equiv p_2 \equiv p_3 \equiv 5 \pmod 8$, such that $p_i$ is a quadratic residue modulo $p_j$ for any $i\neq j$. Prove that, for any prime $p$,...
0
votes
1answer
32 views

How does Fermat's Little Theorem help find primitive roots of unity?

I am asked to find the primitive roots of unity mod 23, and recommended to use Fermat's Little Theorem to help simplify calculation. I know that, once I've found one, I can find all others by finding ...
-1
votes
1answer
58 views

A simple doubt in number theory problem for an even number, [duplicate]

I considered an even number $n\geq 9$, where it is divisible by some positive integer $k$. Also, $k$ does not divide $\frac{n}{2}\ $ (negation of the hypothesis in my prior question). Let $n = kq$. ...