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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3 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 $$ \lfloor x^k \rfloor \equiv m \pmod{n}? $$
0
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0answers
10 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 ...
3
votes
1answer
76 views

Intersection between the sums of the first integers, primes and non primes

Conjecture : $$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\ n\in\Bbb Z\right\} \cap \left\lbrace ...
6
votes
3answers
231 views

Asymptotic density of powers of primes

I'm supposed to compute the asymptotic density of the set \begin{equation} \Pi(x):=\#\{p^k \leq x :p \;prime, k \in \mathbb{N}\} \end{equation} of prime powers less or equal to $x$, that is, compute ...
0
votes
0answers
60 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
56 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 ...
5
votes
3answers
74 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 ...
13
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1answer
256 views

a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction of the square of ramanujan's octic continued fraction which I discovered using certain three term ...
3
votes
2answers
42 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
44 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?
1
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0answers
42 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 ...
2
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1answer
71 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 ...
4
votes
1answer
61 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 $$ \frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2} \qquad \gcd(x,y,z)=1 $$ ...
0
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0answers
22 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 ...
2
votes
1answer
32 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 ...
4
votes
2answers
45 views

Divide a square into different parts

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 geometry, which perhaps yields the shortest, simplest proofs, but other ...
6
votes
1answer
104 views

FRACTRAN for natural numbers

Is there a simple analogue of FRACTRAN that maps a natural number to a natural number, instead of mapping a list of fractions to a natural number? One could use Gödel encoding to translate FRACTRAN ...
5
votes
3answers
409 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
36 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 $\ln(p_{n-1})$?
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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 $$x^{5}+\frac{1}{x^{5}}$$
9
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2answers
459 views
+100

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
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
154 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 ...
2
votes
2answers
81 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 ...
0
votes
0answers
45 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?
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votes
0answers
15 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 ...
3
votes
0answers
89 views
+200

Algorithm to answer existential questions - Reduction

Lemma 1. For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and only if $x$ divides $1$ and ...
7
votes
4answers
2k views

Show that $f(2n)= f(n+1)^2 - f(n-1)^2$

Let $f(n)=f(n-1)+f(n-2)$ be the Fibonacci sequence with $f(0)=0,f(1)=1$. Show that $$f(2n)= f(n+1)^2 - f(n-1)^2.$$ I have tried several different approaches to this problem. Both inducting from the ...
-1
votes
2answers
74 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
68 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 ...
7
votes
1answer
934 views

Why the Riemann hypothesis doesn't imply Goldbach?

I'm interested in number theory, and everyone seems to be saying that "It's all about the Riemann hypothesis (RH)". I started to agree with this, but my question is: Why then doesn't RH imply the ...
1
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1answer
75 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 ...
5
votes
0answers
88 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 $\vartheta_3$ ...
1
vote
2answers
125 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} - ...
0
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0answers
46 views

Proofs needed for observations regarding prime-partitionable numbers.

Below is the definition of a prime-partitionable integer taken from W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1987) 263-272 and is apparently the same in ...
9
votes
1answer
130 views

What do we know about the first occurrences of prime gaps?

Are there any conjectures from which we can infer something about the first occurrences of prime gaps length $n$ and their distribution? I've made an interesting graph of these values to make this ...
128
votes
1answer
7k views

A variation of Fermat's little theorem

Fermat's little theorem states that for $n$ prime, $$ a^n \equiv a \pmod{n}. $$ The values of $n$ for which this holds are the primes and the Carmichael numbers. If we modify the congruence ...
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 ...
0
votes
0answers
73 views

Zero-Sum Partitions of Nonzero Elements of a Ring

In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall almost exclusively discuss finite commutative ...
3
votes
0answers
40 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!
6
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1answer
228 views

Proving $11! + 1$ is prime

Prove that: $$11! + 1$$ is a prime number. Without computing the number (or factorial). Obviously, from Wilson's theorem, a number $n$ is prime if, $$(n-1)! + 1 \equiv 0 \pmod{n}$$ Since $n = ...
1
vote
2answers
38 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 $110$ ...
63
votes
7answers
5k views

How can adding an infinite number of rationals yield an irrational number?

For example how come $\zeta(2)=\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}$. It seems counter intuitive that you can add numbers in $\mathbb{Q}$ and get an irrational number.
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votes
2answers
48 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
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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
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3answers
107 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
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 ...