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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1answer
39 views

A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
4
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6answers
238 views

Sum of an unorthodox infinite series

$ \frac{1}{2^1}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+\cdots $ This is a pretty unorthodox problem, and I'm not quite sure how to simplify it. Could I get a solution? Thanks.
1
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1answer
25 views

Some questions about sub-fields of the field of complex numbers

Given a sub-field $f$ of the field $\mathbb{C}$ of complex numbers, is there a name for the smallest sub-field $F(f)$ of $\mathbb{C}$ such that (1) $F(f)$ contains $f$ as a sub-field and (2) ...
0
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0answers
20 views

Do all complex zeros of $Li_s(z)\,- \, Li_{1-s}(z)$ get the shape $s=\dfrac12 + \dfrac{k \, \pi }{\,\ln(2)}\,i$ when $z \rightarrow 0^{-}$?

This is a subquestion of this question on MO. Numerical evidence strongly suggests that when $z \rightarrow 0^{-}$ the complex zeros that lie in the critical strip $0 \lt \Re(s) < 1$ of: ...
1
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0answers
11 views

Prove that for $a>1$ $\Psi_a( n) \neq \infty$ $\forall a, n\in \mathbb{N}$

Let us consider successive powers of an integer $a$ $(>1)$. Let $\Psi_a(n)$ denote the exponent at which $n$ first occurs in the decimal expression. For example $\Psi_2(2)=1$ and $\Psi_2(3)=5$. ...
0
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2answers
42 views

A number system that is not unique factorization domain

Can anyone present a number system that is not unique factorization domain and is a commutative ring? So I want the case that does not involve polynomials/monomials or some trivial cases.
0
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1answer
28 views

Is there an upper bound (better than Robin's inequality) to the sum of divisors of non-highly-abundant numbers?

I'm looking for $ f(n) $ such that $ \sigma(n) \le f(n) < ne^\gamma \log \log n $, with $ n $ not highly abundant. I'd like a proof as well. I hope the question is well formatted, I'm posting ...
4
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2answers
52 views

Find the 1005th digit after the decimal point expansion of the square root of N.

Let $N$ be the positive integer with $2008$ decimal digits, all of them $1$. That is, $N=1111...1111$, with $2008$ occurrences of the digit $1$. Find the $1005th$ digit after the decimal point ...
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0answers
22 views

Another question/observation about Mersenne numbers and Euler's totient function

This is a follow up to this question Upper bound for Euler's totient function on composite Mersenne numbers and an ongoing project with lots of questions related to Mersenne numbers. I'm sorry if ...
0
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1answer
32 views

How would you show $\pi(x)\log(1-\frac{1}{x}) \sim \frac{1}{\log x}$

How would you show $\pi(x)\log(1-\frac{1}{x}) \sim \frac{1}{\log x}$? Would you use $\lim_{x\to \infty}\frac{\pi(x)\log(1-\frac{1}{x})}{\frac{1}{\log x}} = 1$? and how would you show this? Can you ...
0
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0answers
38 views

Using The Abel Summation formula to calculate $\prod\limits_{p \leq x}(1-\frac{1}{p})$

Using The Abel Summation formula to calculate $\prod\limits_{p \leq x}(1-\frac{1}{p})$ Can anyone give me some hints on how to solve this? I've tried using logs and get \begin{align} ...
0
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0answers
42 views

Which is the best book on Goldbach conjecture research

Is there a book which summarizes the major research results in the past, and current research trends, for the Goldbach conjecture? I know, much progress has been made in Analytic Number theory in ...
0
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0answers
43 views

What is the highest power of the prime p [on hold]

What is the highest power of the prime p dividing :- a) the product 2.4.6....(2n) of the first n even integer b) the product 1.3.5 ... (2n+1) of the first n odd integers
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0answers
30 views

Mersenne numbers with two distinct prime factors

For an integer $k$, denote with $p_k$ the $k$-th prime factor. Let $q$ be an odd prime such that $M_q = 2^q-1$ has exactly two distinct prime factors, say $p_s, p_{s+i}$. What is the largest possible ...
0
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0answers
24 views

Counting/bounding number of relatively prime pairs?

I'm wondering if anyone knows of results counting or bounding the number of relatively prime pairs in two subsets of positive integers. In particular: Given $A = \{a \in \mathbb{Z} | m_1 \leq a \leq ...
2
votes
3answers
74 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
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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
40 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
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1answer
94 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
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3answers
91 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.
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0answers
27 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
150 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
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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
31 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
28 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
27 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 ...
1
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0answers
96 views

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
63 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
76 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
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0answers
52 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
42 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
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1answer
45 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
141 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
34 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
64 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
59 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
46 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
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1answer
65 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
90 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
55 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
113 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
92 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
32 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 ...
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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 ...
21
votes
1answer
163 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
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0answers
33 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
71 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
92 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 ...