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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0answers
57 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
28 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
40 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
43 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
70 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
38 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
71 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
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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 ...
4
votes
1answer
57 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
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1answer
29 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
406 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 $\ln(p_{n-1})$?
0
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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?
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 ...
-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
73 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 ...
0
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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 ...
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 $\vartheta_3$ ...
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 ...
1
vote
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 ...
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
39 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
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2answers
117 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
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 $110$ ...
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
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
-1
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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0answers
30 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 ...
9
votes
1answer
125 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 ...
-4
votes
0answers
40 views

A number theory problem. [on hold]

If $\gcd(a,b) =1$, prove that $\gcd(a-b+bm, a-b+bn) = 1$ where $n= a + bm$.
0
votes
1answer
43 views

Why is the Bernoulli Number $B_1$ sometimes $+ \frac{1}{2} $?

By using the recursive formula, \begin{equation} \sum_{i=0}^{n} \binom{n+1}{i} B_i = n+1 \end{equation} we find $B_1$ to be $\frac{1}{2}$ and not $- \frac{1}{2}$. Why is this?
13
votes
1answer
247 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 ...
4
votes
0answers
41 views

Dividing the whole into a minimal amount of parts to equally distribute it between different groups.

Suppose we have a finite amount of numbers $x_1, x_2, ..., x_n$ ($x_i\in\mathbb{N}$) and an object that should be divided into parts in such a way that it can be without further dividing distributed ...
0
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1answer
30 views

question about first occurring prime gaps

If a prime gap $g(p)$ is the first occurring prime gap of it's size, does this imply that it is also the largest gap below $p$? In other words, is the set of first occurring prime gaps contained ...
4
votes
1answer
33 views

Pair of Circles Intersect

If $S$ is a collection of circles s.t. for each point $c$ on the x-axis there is a circle in $S$ passing through the point $(c,0)$ and at the same time has the x-axis as a tangent to the circle at ...
3
votes
0answers
85 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 ...
5
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0answers
85 views

Are there papers or books that explain why Bernhard Riemann believed that his hypothesis is true?

I would like to know what are the mathematical reasons for which Bernhard Riemann believed that his hypothesis is true, and I would like to know if those mathematical reasons were cited in his ...
3
votes
2answers
52 views

Permutations minus Transpositions

I want a formula that allows me to find all the permutations in $S_n$ (which is the set of all the integers from 1 to $n$) which don't contain a transposition. Attempt: Lets call $g(n)$ the ...
0
votes
1answer
67 views

algebra question.. [on hold]

If $f : \mathbb{R}\rightarrow \mathbb{R}$, and $f(x)=\frac{2}{4^{x}+2}$ Find the value of $$f\left [ \frac{1}{11} \right ]+f\left [ \frac{2}{11} \right ]+ \cdots +f\left [ \frac{10}{11} \right ]$$
2
votes
2answers
52 views

Terms of a certain recurrence

Let $a_1, a_2\dots $ be a sequence of reals such that $a_1 = a_2 = 1$, and $$a_{n + 2} = \frac{a_{n + 1}^3 + 1}{a_n}$$ for $n \ge 1$. It appears to be the case that all of these values are integers. ...
3
votes
1answer
79 views

Frobenius automorphism and non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$

For completeness I give some definitions. Let $p$ a prime number and consider $V$ a 2-dimensional $\mathbb{F}_p$-vector space. Consider $k$ a sub-algebra of $\mathrm{End}(V)$ that is a field of $p^2$ ...
3
votes
2answers
63 views

Indexes of prime Fibonacci numbers

I found this on Mathworld, but I can't seem to find any proof, either on StackExchange, nor any other site: Why do all Fibonacci primes, except for $F_4=3$, have prime indexes (with $F_0=0$)? My ...
1
vote
1answer
28 views

Fermat primality test and Fermat pseudoprime

What is the difference between Fermat primality test and Fermat pseudoprime?Can anyone explain me how we use them ?
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 ...