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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2
votes
3answers
99 views

remainder when 67896789…(300 digits) divided by 999

What is the remainder when 678967896789... (300 digits)is divided by 999? i tried to divide it manually to find some pattern in remainder. But was getting bit lengthy. so please suggest me some short ...
0
votes
0answers
23 views

Is there an upper bound to the sum of the divisors of a non-highly abundant number which is better than Robin's inequality?

Since it is known that all large odd integers satisfy Robin's inequality and that the least counterexample to it must be superabundant, I was wondering if there is a function $\ f(n) $ such that $\ ...
4
votes
1answer
43 views

If $G/H$ and $G$ are connected linear algebraic groups must $H$ also be connected?

Let $k$ be a perfect field (e.g of characteristic zero) and let $G$ and $H$ be linear algebraic groups over $k$, with $H$ a normal subgroup of $G$. If both $G$ and $G/H$ are connected, must $H$ ...
2
votes
1answer
98 views

When does $2^n+n \mid 8^n+n$?

How to find all positive integers $n$ such that $2^n+n$ divides $8^n+n$ ?
4
votes
3answers
107 views

Find all positive integers satisfying $\frac{2^n+1}{n^2} =k $

Find all positive integers satisfying $$\frac{2^n+1}{n^2} =k $$ where $k$ is a integer. I can't just come up with a solution.
-3
votes
0answers
33 views

show this sets $S=\{\lfloor n\pi \rfloor|n=0,1,2,\cdots\}$ contains arbitrarily finite long arithmetic progressions [duplicate]

Question: Define sets $$S=\{\lfloor n\pi \rfloor|n=0,1,2,\cdots\}$$ show that: the sets $S$ contains arbitrarily finite long arithmetic progressions,But not contains infnite long ...
8
votes
2answers
159 views

Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
1
vote
0answers
40 views

What is a “Diophantine number”? [duplicate]

In this phrase, excerpted from Terry Tao's blog, "$\alpha$ being ... very far from rational (a Diophantine number), ..." what is a "Diophantine number"? He contrasts it with a Liouville number. ...
2
votes
1answer
49 views

Is every positive integer the sum of at most 8 pentatope numbers?

Is every positive integer the sum of at most 8 pentatope numbers ? See : http://en.wikipedia.org/wiki/Pentatope_number I saw this conjecture here : ...
0
votes
1answer
36 views

Solution to Diophantine equation with constraint.

solve the following equation over $z_x,z_y$ \begin{align} &az_x=bz_y\\ &\text{s.t. } a,b,z_x,z_y \in \mathbb{Z} \text{ and } 1 \le z_x \le N \text{ and } 1 \le z_y \le N \end{align} How ...
0
votes
1answer
28 views

Count ways to sit men women in row of size K

Suppose we are given N men and M women.They are to sit in a row of size K such that no two women sit next to each other.What are the number of ways. Like if suppose their are 3 men and 2 women and ...
3
votes
1answer
245 views
+50

Prove or disprove that there exists a unique positive integer sequence $\{a_{n}\}$ satisfying a condition

Question: Prove or disprove: there exists a unique positive integer sequence $\{a_{n}\}$ satisfying the following condition: $\forall m\in N^{+}$, there exists a unique integer sequence ...
4
votes
3answers
70 views

how find $\sum_{k \in A} \frac{1}{k-1} $ for $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $

If $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $, then how find $\sum_{k \in A} \frac{1}{k-1} $?
3
votes
1answer
78 views

Remainder on dividing $10^{n} + 10^{n-1} + … + 10^{1} + 10^{0}$ by x

Given a positive integer $n$, consider the number $y=10^{n}+10^{n-1}+$$...+ 10^{1}+10^{0}$. I need to find the remainder when $y$ is divided by a natural number $x$. e.g. $111111$ $\%$ $2123$ = ...
1
vote
0answers
53 views

Comparing a number with a line of power

How do you compare which is bigger (or maybe equal), LHS or RHS, in $$a \sim b_1^{b_2^{.^{.^{.^{b_n}}}}}$$ given $a$ and $b_i$, $1 \leq i \leq n$, are non-negative integers (also could be big)? The ...
0
votes
1answer
44 views

Under what conditions can we obtain $a \equiv 1 \pmod{mn}$ from $a \equiv b \pmod{m}$ and $b \equiv 1 \pmod{mn}$?

If $a \equiv b \pmod{m}$ and $b \equiv 1 \pmod{mn}$, are there any conditions under which we can conclude that $a \equiv 1 \pmod{mn}$? Here $m$ and $n$ are any integers; $a$ and $b$ are both coprime ...
1
vote
1answer
48 views

Residue class of a huge repunit modulus a huge number

Given a number with only 1: X = 1111...1 (N times 1 in total), and another number M, I want ...
2
votes
0answers
144 views

$e=1$ in Theorem 30 from Marcus book “number fields”

Theorem 30 in Marcus book states that, if $p\in\mathbb Z$ is an odd prime and $q$ is a prime $\neq p$, then, fixing $d$ as a divisor of $p-1$ we have that $q$ is a $d$-th power $\operatorname{mod}q$ ...
0
votes
1answer
36 views

How many distinct lists of 14 integers $L=\{v_1,\ldots ,v_{14}\}$ exist satisfying $v_i \geq v_{i+1}\geq 0$ and $\sum _{i=1}^{14}(v_i) \leq 54$

I am trying to solve the following problem: I have an ordered list of integers $L = \{v_1,\ldots ,v_{14}\}$ with fourteen elements, satisfying the following two conditions: $v_i \geq v_{i+1}\geq 0$ ...
4
votes
0answers
66 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
0
votes
1answer
60 views

Count numbers with prime digit

Given a number N I need to find the count of the numbers that have atleast one prime digit (2,3,5 or 7) in it. Now N can be upto 10^18.What is the best approach to solve this problem. Example : Let ...
2
votes
1answer
47 views

Is the law of large numbers reflected in SO reputation system?

I have recently encountered an interesting phenomenon on SO reputation system: Let $f(n)$ denote the current score of the $n$th best user. A sample that I collected at a given moment: $f(k)=k ...
2
votes
1answer
30 views

Finding the lowest number (or an upper bound to the lowest number) not congruent to a set of moduli

Note: if finding x is not possible, an upper bound, where there must be at least one number less than said number which is not congruent to the set, would be helpful. The set: For my purposes, the ...
3
votes
1answer
71 views

How can the smallest set of integers be generated such that the sums cover a given set?

I have a set of positive integers S. I want to generate a set of positive integers T such that every member of S is the sum of some combination of members from T. I am looking for the smallest ...
9
votes
1answer
101 views

For what integers $n$ is this divisibility statement true?

The statement being $$n^2 + 2 \mid 2014n + 2$$ The answer is $n = -2, 0, 1, 2014$. Don't know how to arrive at this answer without using comp sci. (Using the compsci answer, we can restrict the ...
2
votes
1answer
56 views

Hard Simultaneous Diophantine Equations

Find all positive integers $a,b,c,d,e,f$ such that : $de^2=ab^2+1$ and $df^2=ac^2+1$. I tried subtracting them, it factors quite nicely. But after that, haven't a clue. I'm not sure if it's even ...
2
votes
1answer
117 views

Are there any “obviously” true propositions in number theory?

After all efforts spent on wrong proofs of famous number theory conjectures and theorems like Goldbach's or Fermat's last theorem, could one find some simple statements (might be correct ones) whose ...
2
votes
3answers
97 views

Irrational number “test”?

Suppose we have a finite quantity $a$, which we would like to prove to be irrational, supposing that it is indeed irrational. Then, would it be enough to show that ...
0
votes
1answer
33 views

Counting points in/on cuboid

Given a cuboid that extend in x,y,z axis such that |x|≤N, |y|≤N, |z|≤N where N is given and can have value up to 10^9.Now a shooter is standing at origin (0,0,0).He need to shoot on any of the ...
1
vote
2answers
46 views

$a^2-b^2 = k$, $ab = l$ for fixed integer $k,l$ when $a,b$ are both integers

Let us fix integers $k,l$. Let all numbers be integers. Now we want integer $a,b$ to satisfy: $$a^2-b^2 = k, \,\,\,2ab = l.$$ We want to maximize the number of possible $(a,b)$. In order to do ...
22
votes
1answer
226 views

Which is greater, $20 \uparrow\uparrow\uparrow\uparrow 20$ or $4 \uparrow\uparrow\uparrow\uparrow\uparrow 4$?

This past Wednesday's What-If had this image at the bottom: In particular, I am interested in $20 \uparrow\uparrow\uparrow\uparrow 20$. I immediately thought of Graham's Number, but clearly that ...
3
votes
0answers
34 views

Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
2
votes
2answers
52 views

Can there ever be infinite number of tuples of $(a,b,c,d)$ such that $ac-bd = k$ and $ad+bc = l$ for fixed $k,l$?

Suppose, for now, that all numbers are real numbers. Let us fix numbers $k,l$. Then can there ever be infinite number of tuples of $(a,b,c,d)$ such that $ac-bd =k$, $ad+bc = l$ for some $k$ and $l$? ...
5
votes
1answer
75 views

On the square coeffecients of a modular form

Let $k\in \mathbb{N}$. Let $f\in M_k(\Gamma_0(N),\chi)$ be a modular form of weight $k$ on $\Gamma_0(N)$ with a Dirichlet character $\chi$. If $f$ has a Fourier expansion of the form $$ ...
0
votes
2answers
94 views

Modulus of large powers

Given an array of N integers where $2 ≤ N ≤ 2×10^5$ and each element in array is less than $10^{16}$. Now I am given a variable $X$ that can also go up to $10^{16}$. We need to find if $X \mid ...
0
votes
2answers
79 views

Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$

Let $a,b,c$ be co-prime integers $>2$ . Find all non-trivial triplets $(a,b,c)$ such that $ a+b^{d}\equiv0\pmod c $ for all $d>0$.
0
votes
1answer
61 views

Can anyone recommend an easy to read algebraic number theory book?

Can anyone recommend an easy to read algebraic number theory book ? I prefer a book with good examples. (hints or answers to selected questions if possible. Not sure if it is possible for a book of ...
12
votes
2answers
281 views

Why are people more interested in the Riemann hypothesis than Goldbach's conjecture? [closed]

One of my friends, a math professor, told me almost every one of his colleagues (in the math department) had attempted to prove the Riemann hypothesis at some point in their life (maybe secretly). ...
0
votes
1answer
37 views

Convergence of $\sum_{n=1}^{\infty} n$ and integral test [duplicate]

I have read online, that one can show that $\sum_{n=1}^{\infty}n = -\frac{1}{12}$. But isn't this a Riemann Series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p=-1$. And if so, can't ...
3
votes
2answers
55 views

Which natural numbers can be represented as a sum of natural numbers raised to different powers?

Waring's problem asks about natural numbers that can be represented as a sum of natural numbers all raised to the same power $k$. I'm wondering which natural numbers can be represented as a sum of ...
4
votes
0answers
154 views

Prove modular equivalence related to carmichael function.

Statement if $n=p_1p_2p_3\cdots p_r$ (prime numbers are distinct) and $2 \le a \le n-1$ then prove that $$a^{k\lambda} \equiv \left(1+\sum_{p_i\mid\gcd(a,n)}\left(\dfrac{n}{p_i} \right) ...
1
vote
3answers
53 views

Rational numbers and periodic decimal representation

I'm trying to prove that a number is rational if and only if it has an eventually periodic decimal expansion. One part is simple; without loss of generality we consider $q=0.\overline{d_1\dots d_k},$ ...
3
votes
3answers
105 views

Upper bound for Euler's totient function on composite numbers

I've seen before the general bound $\phi(n) \leq n - n^{1/2}$ for composite $n$. Can this bound be improved at least for those $n$ when we don't have equality above? Say could we possibly have at ...
4
votes
1answer
79 views

Factoring numbers

I was given the following problem: Prove that 767, 76767, 7676767 ... are all composite. Making a sequence $a(n)$ = {767, 76767, 7676767, ...} you can show that $a(3k+1)$ is divisble by 13, ...
10
votes
2answers
202 views

Cubic polynomial equal to a cube

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers. $$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...
0
votes
0answers
44 views

Diophantine Equations involving cubes

I'm doing some number theory research and I came across these two Diophantine equations (created under my own transformations): $$y^3 = ax^3 + bx$$ (where $a$ and $b$ are parameters) $$z^3 = x^2 + ...
1
vote
2answers
61 views

Why attempt to find 'b' in golden ratio ends up with odd results?

Imagine that you have a line segment which is divided into larger and smaller segment. You want to find a golden ratio for the longer segment $a$, and shorter one $b$. As you know the phi number which ...
2
votes
1answer
26 views

Using continued fractions to well-approximate a quadratic form?

Continued fractions are the "best rational approximation" of other numbers. For a real number $\alpha$ the continued fraction algorithm produces a sequence of integers $\alpha = [a_1, a_2, \dots, ...
6
votes
1answer
64 views

When does $(a^p-1)/(a-1)$ have a $p$th power factor?

Today I noted that $(18^3-1)/(18-1) = 343 = 7^3$, and that there are no other solutions to the equation $(a^3-1)/(a-1) = b^3$ with $b \le 100000$. There are, however, many solutions to the equation $$ ...
2
votes
0answers
28 views

floor function inequality $\frac{3c-3n-3}{2n+3} \geq \lfloor\frac{3c-3n}{2n+3}\rfloor$

I want to prove following statement: $$\forall K\in \mathbb R \exists c\in \mathbb N c\geq K s.t.\forall n\in \left\{1,2,...,c-1\right\}: \frac{3c-3n-3}{2n+3} \geq \lfloor\frac{3c-3n}{2n+3}\rfloor$$ ...