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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0
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0answers
14 views

If $(a_n)$ is increasing and $a_n^{1/c^n}\to\infty$, then $\sum\frac1{a_n}$ is irrational?

I $\DeclareMathOperator\lcm{lcm}$am trying to generalise the result from this question: If $(a_n)$ is increasing and $\lim_{n \to \infty} a_n^{1/2^n} = \infty$, show that $\sum_{k=1}^{\infty} 1/{a_n}$ ...
0
votes
1answer
23 views

What does Residue multiplication mean?

Suppose $ a, b, c, n \in \mathbb{Z}, \qquad$ where $n>0, \qquad$ then $a\cdot b \texttt{ mod } n = c$ is called modular multiplication. The article that I am reading mentions Modular and Residue ...
2
votes
1answer
34 views

Longest sequence of primes with no difference greater than $2n$

Let $N\ge 1$ be a natural number. The object is to find the longest possible sequence of prime numbers $p_1<p_2<...<p_n$ such that $p_{i+1}-p_i\leq 2N$ for $i=1,...,n-1$. In other words, ...
6
votes
1answer
47 views

Why does $x^2+47y^2 = z^5$ involve solvable quintics?

This is related to the post on $x^2+ny^2=z^k$. In response to my answer on, $$x^2+47y^2 = z^3\tag1$$ where $z$ is not of form $p^2+nq^2$, Will Jagy provided one for, $$x^2+47y^2 = z^5\tag2$$ as, ...
0
votes
0answers
11 views

To use topology and Riemann surface in number theory.

I would like to learn some rudiment topology and Riemann surface in order to apply in number theory. I already know some algebraic topology, like covering space and fundamental group, singular ...
0
votes
0answers
14 views

Grothendieck-Lefschetz fixed point theorem for nonconstant sheaves?

Let $X$ be a variety over ${\bf F}_q$. The Grothendieck-Lefschetz fixed point formula gives $$|X({\bf F}_q)|=\sum_i(-1)^i\text{Tr}(\text{Frob},H^i_c(X,\bf{Q}_\ell))$$ for the constant $\ell$-adic ...
0
votes
1answer
22 views

How to show that $\prod^{p-1}\limits_{j=0} (x+\eta^jy)=x^p+y^p$

How to show that $\prod^{p-1}\limits_{j=0} (x+\eta^jy)=x^p+y^p$, where $p$ is an odd prime, $\eta$ is $p$-th roots of unity and $x,y$ are integers. It could be reduced to the form ...
5
votes
6answers
173 views

$A+B+C=2149$, Find $A$

In the following form of odd numbers If the numbers taken from the form where $A+B+C=2149$ Find $A$ any help will be appreciate it, thanks.
2
votes
0answers
21 views

Minimizing a floor expression

Consider the expression $$ax - b\left\lfloor\frac{cx}{m}\right\rfloor$$ Variables $a, b, c, m$ are positive integers (all of which are known), and $x$ is an unknown integer. The bounds on $x$ are ...
5
votes
5answers
111 views

$p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.

I want to prove that for $p,q,r$ different primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational. Is the following proof correct? If $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is rational, then ...
2
votes
1answer
44 views

Existence of a map

Does there exist a map $\phi \colon \mathbb N\cup \{0\} \rightarrow \mathbb N\cup \{0\}$ that holds the following property? $$\phi (ab) = \phi(a)+ \phi(b)$$ If they do what do they ...
1
vote
1answer
61 views

Number of zeros at the end of $k!$

For how many positive integer $k$ does the ordinary decimal representation of the integer $k\text { ! }$ end in exactly $99$ zeros ? By inspection I found that $400\text{ !}$ end in exactly $99$ ...
0
votes
1answer
27 views

Tight bounds for Bowers array notation

This link http://googology.wikia.com/wiki/Array_notation shows the definition of bowers linear array notation and the approximation $$\{n,a+1,b+1,c+1,d+1,...\}\ \approx f^a_{...+\omega^2d+\omega ...
-2
votes
0answers
27 views

What the nontrivial zeros have to do exactly with the distribution of the primes? [on hold]

Why is it very hard to prove Riemann hypothesis, I mean I know that if I can prove that there are no nontrivial zeros outside the critical line x = 1/2, then RH will be proven to be true, but what it ...
4
votes
1answer
49 views

$\lfloor x^k \rfloor \equiv m \pmod{n}$ with $x$ irrational

Let $x>1$ be an irrational number, and $n$ a positive integer. Is it true that, for each integer $m$, there exists an integer $k$ such that $$ \lfloor x^k \rfloor \equiv m \pmod{n}? $$
-1
votes
0answers
16 views

How can I use diophantine approximation to find a real number?

I have been told that the following question can be solved using Diophantine approximation, but I cannot find a way to solve it. I have no prior knowledge of Diophantine approximation and so I ...
0
votes
0answers
64 views

Is 1 really equal to 0.99999999… [duplicate]

I heard a few times that 1 is equal to $0.9999 \dots$ (infinite nines). I know that the limit of this is actually 1, but does that that the equivalency hold here? Can't we argue that $1 - 0.99999 ...
1
vote
3answers
61 views

Number theory problem and Diophantine Equations

Suppose $m^3=n^4-4$ where $m,n \in \mathbb Z$. a) Show that $m$ cannot be even if $n$ is odd. b) Show that $m$ and $n$ cannot both be even. c) By considering the prime factors of ...
3
votes
2answers
45 views

Smallest multiplier to make a rational number whole

This might be a really stupid question. For a given rational number q, is there a simple way of finding the smallest natural number n such that qn is a natural number?
-2
votes
1answer
46 views

Solve the equation below [on hold]

Solve the equation $$\tan(\cos^{-1}\sqrt{x})=2^{\log_{4}x}.$$ I have no idea where I have to start; it's a little hard for me. So any help?
2
votes
1answer
75 views

Find integer solutions equation of ${ x }_{ 1 }^{ 4 }+{ { x }_{ 2 }^{ 4 }+ }{ x }_{ 3 }^{ 4 }+…+{ x }_{ 14 }^{ 4 }=1599 $

I tried to solve this equation,but can't end up $${ x }_{ 1 }^{ 4 }+{ { x }_{ 2 }^{ 4 }+ }{ x }_{ 3 }^{ 4 }+...+{ x }_{ 14 }^{ 4 }=1599$$ My work: Consider arbitrary $x_{ i }=2k,\quad \forall ...
1
vote
0answers
53 views

Minimum Cake Cutting for a Party

You are organizing a party. However, the number of guests to attend your party can be anything from $a_1$, $a_2$, $\ldots$, $a_n$, where the $a_i$'s are positive integers. You want to be ...
5
votes
3answers
80 views

How to prove there are no solutions to $a^2 - 223 b^2 = -3$.

As the title suggests, I'm trying to prove that there are no solutions to $a^2 - 223b^2 = -3$ (with $a,b\in \mathbb{Z}$). Ordinarily, taking both sides $\mod n$ for some clever choice of $n$ proves ...
0
votes
0answers
29 views

Minimizing a sum given variables

I have this expression $$ax - b\left\lfloor\frac{cx}{m}\right\rfloor$$ Variables $a, b, c, m$ are known/given positive integers, and $x$ is an unknown integer with bounds $1 \leq x \leq m-1$. I ...
4
votes
1answer
63 views

Solution to Diophantine equation $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2} $

I have to prove the following, but I don't know how to start. The only solutions in positive integers of the equation $$ \frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2} \qquad \gcd(x,y,z)=1 $$ ...
2
votes
1answer
43 views

Two disjoint number fields $K$, $L$ such that $(\mathrm{disc}({\cal O}_K), \mathrm{disc}({\cal O}_L))\neq 1$ but ${\cal O}_L{\cal O}_K={\cal O}_{KL}$

I know that if two disjoint number field $K$, $L$ are such that $(\operatorname{disc}(\mathcal{O}_K), \operatorname{disc}(\mathcal{O}_L))= 1$ then $\mathcal{O}_L\mathcal{O}_K=\mathcal{O}_{KL}$. It is ...
5
votes
3answers
418 views

Problem Solving Positive Integers

This is a very interesting word problem that I came across in an old textbook of mine. So I know the maximum value of the HCF has to be a factor of $540$ and mayhaps the Euclidean Algorithm, but other ...
-1
votes
1answer
40 views

Question on occurrences of prime gaps [on hold]

Why is the number of times a prime gap $p_{n} - p_{n-1}$ is above $\ln(p_{n-1})$ always the same as the number of times it occurs below $\ln(p_{n-1})$?
0
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0answers
48 views

Solutions to ax = by mod m?

Given congruence $ax = by \bmod m $ for known integers $a,b,m$, with $m $ composite, can this relation be simplified or solved?
3
votes
2answers
56 views

If $x^a \equiv x^b \bmod p$, what can we say about $a$ and $b$?

If $x^a \equiv x^b \bmod p$, what can we say about $a$ and $b$, for $p$ prime? Is there any way to show the relationship between $a$ and $b$ specifically? It doesn't seem to be the case that $ a ...
5
votes
2answers
160 views

Prove that every integer $n\geq 7$ can be expressed as a sum of distinct primes.

My teacher said to use Bertrand's postulate and I have tried this for so long and I seem to go nowhere. Help would be appreciated. EDIT: Here's what I've done in my proof so far (I need help ...
-3
votes
4answers
179 views

Find the value of the question below [on hold]

If $x^{3}+\frac{1}{x^{3}}=14$ Find the value of $$x^{6}+\frac{1}{x^{6}}$$ Original Question: If $x^{2}+\frac{1}{x^{2}}=14$ Find the value of $$x^{5}+\frac{1}{x^{5}}$$
-1
votes
2answers
77 views

Evaluate the infinite radical expression $2\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2 \cdots}}}}$ [on hold]

Find the value of $$2\sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2 \cdots}}}} .$$ I really don't know where I start, so any help will be appreciated.
0
votes
5answers
74 views

Deriving Euler's theorem from Fermat's little theorem

I would like to know why $a^p \equiv a \pmod p$ is the same as $a^{p-1} \equiv 1 \pmod p$, and also how Fermat's little theorem can be used to derive Euler's theorem, or vice versa. Please keep in ...
0
votes
0answers
16 views

Binary solutions of multivariate polynomial system in special (factored) form.

In my personal research I've run into a system of multivariate polynomials (with coefficients in a field). I am aware that there is no polynomial time algorithm (in the number of indeterminates) for ...
5
votes
0answers
95 views

The number $\sum\limits_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$ is transcendental

Prove that the number: $$\sum_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$$ is transcendental. I don't have a direct proof but a round one. The series can be expressed in terms of $\vartheta_3$ ...
2
votes
2answers
84 views

$\zeta(2n)$ proof [duplicate]

Can anybody pass me on a good source to see the steps in proving, \begin{equation} \zeta(2n) = \frac{(-1)^{k-1}B_2k (2 \pi)^{2k}}{2(2k)!} \end{equation} I know how we start by looking at the product ...
1
vote
1answer
80 views

Permutations of the elements of $\mathbb Z_p$

Note Added by Robert Lewis, 2 August 2015 3:04 PM PST in an attempt to provide background, motivation, and other context for this engaging problem: This problem essentially asks for a method of ...
0
votes
1answer
36 views

What is the significance to our number and degrees systems? [duplicate]

I saw this video recently and it suggests that there is some "magical" reason that there are 360 degrees in a circle and that it is also connected with our number system. My question is: How did we ...
3
votes
0answers
41 views

$F[t]$ has undecidable positive existential theory in the language $\{+, \cdot , 0, 1, t\}$

Consider the ring $F[t, t^{-1}]$ (the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$). Theorem 1. Assume that the characteristic of $F$ is zero. Then the existentia theory ...
2
votes
1answer
41 views

Number Theory Problem involving fractional part of a number

If $x = ( 9 + 4 \sqrt {5} )^{48}$ where $x = [x] + f$, where $[x]$ is he integral part of $x$ , and $x$ is its fractional part How do I go about finding the value of $x(1-f)$ ? Thanks!
1
vote
2answers
139 views

How come $\ n\ $ always divides at least one of the item of the sequence?

Given positive integer$\ \displaystyle n,\ $ the sequence is: $\displaystyle 2^n$ $\displaystyle 2^n - 2^{n-1}$ $\displaystyle 2^n - 2^{n-1} + 2^{n-2}$ $\displaystyle 2^n - 2^{n-1} + 2^{n-2} - ...
1
vote
2answers
40 views

To calculate the remainder of (111…) + (222…) + (333…) + (444…) + (555…) + (666…) +(777…) by 37

To Evaluate the remainder Question: $ (111...) + (222...) + (333...) + (444...) + (555...) + (666...) +(777...)$ mod $37$ In each bracket, the single digit $(1, 2, 3, ..., 7)$ is written $110$ ...
0
votes
1answer
20 views

something similar to the Bézout's identity, but with three integers.

There are three positive integers,not all equal. And their greatest common divisor is 1. We can perform this operation on them: choose two not equal integer $a,b(a<b)$ from them, and then ...
3
votes
3answers
111 views

Why are $e$ and $\pi$ believed to be normal?

I've found that affirmation in several sources, but I can't think of an obvious reason.
2
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0answers
21 views

A modified Shapiro's Tauberian theorem? Proof or counterexample

Let $\{a(n)\}$ be a nonnegative sequence such that $$\sum_{n\leq x}a(n)[x/n^{2}]=x^{2}\log x + O(x^{2})$$ for all $x\geq 1$, where $[y]$ denotes the greatest integer $\leq y$. Is true that the ...
-7
votes
2answers
46 views

fermat's little theorem prove [on hold]

Prove that the third number of fermat's is prime? any help with the prove ? I meant prove that $257$ is prime
-1
votes
2answers
50 views

Circular table problem

I've looked other questions that might help solve my problem, but haven't found any people who've used my method to solve it. The problem goes like this: Suppose there are 7 men and 5 women, and they ...
0
votes
0answers
31 views

Modified arithmetical functions from a modified Möbius function

In this post when $n>1$ we assume that $n=\prod_{k=1}^{\omega(n)}p_{n}^{e_{k}}$ its factorization in prime powers, where $\omega (n)$ is the number of distinct primes. It is well known the ...
2
votes
0answers
28 views

proof of chinese remainder theorem $x=a_1M_1y_1+…+a_nM_ny_n$?

I can't understand the proof of Chinese Remainder Theorem let $x ≡ a_1 (\text{mod }m_1 ),$ $x ≡ a_2 (\text{mod }m_2 ),$ · · · $x ≡ a_n (\text{mod }m_n )$ such that $m_1,m_2,...,m_n$ are relatively ...