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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how many jelly beans did each girl have at first?

Martha and Mary had 375 jelly beans in all. After Mary ate 24 jelly beans and Martha ate 1/7 of her jelly beans, they each had the same number of jelly beans left. How many jelly beans did each girl ...
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2answers
34 views

Collatz conjecture: Largest number in sequence with starting number n

This question is inspired by a CS course, and it only tangentially relates to the actual content of the exercise. Say in a hailstone sequence (Collatz conjecture) you start with a number n. For any ...
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18 views

All possible number combinations in decimal represenation of irrational numbers?

This question is directly inspired by "Does Pi contain all possible number combinations?". I would like to state firstly for the record that I have no serious number theory education. I think I ...
2
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1answer
22 views

Asymptotic formula for sums of powers of reciprocals of primes

Is there an explicit asymptotic formula, in terms of $\alpha$, for the expression $$\displaystyle \sum_{p \leq x} \frac{1}{p^\alpha}$$ for $0 < \alpha < 1$? The case $\alpha = 1$ is supplied by ...
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0answers
12 views

On a congruence for the number of finite topologies

I am making search about "On a congruence for the number of finite topologies". I have found a paper. I guess it is written in Russian. How can I find English version of this paper ? I am also ...
2
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1answer
21 views

Origin of period function model of primes

There is a web page attributed to Omar Pol, "Sobre el patrón de los números primos: Determinación geométrica de los números primos y perfectos." ("On the pattern of primes: Geometric Determination of ...
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1answer
30 views

What books do you recommend on mathematics behind cryptography?

I am currently reading the Book Understanding Cryptography from Cristof Paar. I am enjoying the book but i don't like to scratch the surface when it comes to cryptography. I would like do dig a little ...
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0answers
63 views

Are $121$ and $400$ the only perfect squares of the form $\sum\limits_{k=0}^{n}p^k$?

I've been looking for perfect squares that can be represented as $\sum\limits_{k=0}^{n}p^k$. Of course, both $n$ and $p$ should be natural numbers larger than $1$. Searching up to $n=100$ and ...
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0answers
13 views

How to expand powers of multiple pairwise commuting elements in a group [on hold]

Let (G, $\ast$) be a group and let n $\in\aleph$. Prove that if $g_1,...,g_k\in G, k\in\aleph$ are pairwise commuting elements of G, then $(g_1\ast...\ast g_k)^n$=$g_1^n\ast ...\ast g_k^n$
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1answer
17 views

Proofed: Every number in the sequence of powers of 2 have $phi = 1/2 * 2^x$

I want to know if it's proofed, that every number which is in the number sequence of the powers of $2$ has an $\phi$ of $\frac12x$.
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1answer
25 views

Induction proof involving Euler Totient Function

Let $\varphi$ be the Euler totient function Qi) show that if $r$ is a power of a prime number then $\sum_{d|r} \varphi(d) = r.$ Qii) Show that if $n \geq 2$ then there is a decomposition of n as a ...
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0answers
14 views

Understanding how to estimate $\pi(x)$ based on Paul Erdos's proof of Bertrand's Postulate

I am reading the 4th Edition of Proofs from the Book. I am not clear on how the proof behind Bertrand's postulate leads to the following statement on page 10 (of my edition): From (2) one can ...
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0answers
13 views

Nebentypus of contragredient representation

Let $k$ be a local non-archimedean field with ring of integers $\cal O$ and maximal ideal $\frak p$. Let $\pi$ be an irreducible admissible $\infty$-dimensional representation of $\text{GL}_2(k)$ ...
4
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1answer
48 views

The number of divisors of a number whose sum of divisors is a perfect square

Let $n$ denote a non-prime whose sum of divisors is a perfect square. I have noticed a few surprising facts on the number of divisors of $n$: It is either prime or semi-prime or $27$ in all cases ...
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0answers
26 views

Clever use of Pell's equation

Find infinitely many triples $(a,b,c)$ of positive integers such that $a,b,c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$are perfect squares. The solution is: Consider the ...
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0answers
24 views

Strict total ordering

I'm not able to understand how the below relation is example of "strict total order". Consider a set $X = 2^Y$ where $Y = \{1,2,3,4,5,6,7,8,9\}$. The expected order of $X$ is for all $x, y$ ...
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0answers
54 views

Are there infinitely many $N^3$ (especially for prime $N$) that cannot be expressed as a sum of three positive cubes?"

Well few days ago i asked a question on perfect numbers and Tito Piezas III answered the question in a very intriguing way which has helped me to get a lead on it.But his answer and perfect numbers ...
3
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1answer
38 views

Two question about ${\mathbb Q}(\alpha)$ for $\alpha$ := $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$

Let $\alpha$ = $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$ with $n$ not divisible by $3$. Prove that $[{\mathbb Q}(\alpha) : {\mathbb Q}] = n(n + 3)$. Conclude that $\alpha$ is constructible if and only if $n = ...
3
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1answer
75 views

Trying to understand a step in Ramanujan's Proof of Bertrand's Postulate regarding the gamma function

My question relates to this step in the proof here: But it is easy to see that $$\log \Gamma(x)-2\log\Gamma(\frac12x+\frac12) \le \log\left\lfloor ...
2
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1answer
61 views

Multiples of 3 and 5. [on hold]

If we have the Tartaglia(Pascal) triangle in every row which numers are multiples of 3 which are even and which are multiples of 5?
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1answer
20 views

Understanding Bézout's identity

I'm trying to understand a proof of Bézout's identity ($gcd(a,b)=$ smallest linear combination of $a$ and $b$), and I'm having some trouble following the last step. The proof goes by: Let $m=sa+tb$ ...
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2answers
33 views

Is there a subset of natural numbers with a special property

Let set $A$ be an infinite big subset of the set $\mathbb{N}$ (set of natural numbers),it is not equal to $\mathbb{N}$ and it has the following property: For every $a$ that is not from the set $A$ ...
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1answer
25 views

How to solve second order congruence equation if modulo is not a prime number

the equation is $x^2 = 57 \pmod{64}$ I know how to solve equations like (*) $ax^2 +bx +c = 0 \pmod{p}$, where $p$ is prime and i know all the definitions for like Legendre's Symbol and all of the ...
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1answer
12 views

Partition of fractional parts where each sum of them has to be at least 1

Let $ a_1,\ldots,a_t \in \mathbb{Q} \setminus \mathbb{Z} $ be with $ \sum_{i=1}^t \lbrace a_i \rbrace \in \left[k,k+1\right) $ for some $ k \in \mathbb{N} $ with $ k \ge 4 $. Here $ \lbrace x \rbrace ...
0
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1answer
138 views

Selfy number couldn't exist?

For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$, and $l(x)$ denote the number of digits of $x$ (in base $10$). Now given a positive integer $n \ge 2$, it seems that there ...
3
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1answer
35 views

Integer solutions to a two variable equation.

For $m, n \in \mathbb{Z}$, show the only integer solutions to $f(m,n) = \displaystyle \frac{3^m(2^n+1)-2^{m+n}}{2^{m+n}-3^{m+1}}$ are $f(1, 2) = -7$, $f(0, 1) = -1$, and $f(0, 2) = 1$. More ...
5
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1answer
154 views

Can the cube of every perfect number be written as the sum of three cubes?

I found an amazing conjecture: the cube of every perfect number can be written as the sum of three positive cubes. The equation is $$x^3+y^3+z^3=\sigma^3$$ where $\sigma$ is a perfectnumber (well it ...
5
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0answers
61 views

Which prime gaps are known to exist [duplicate]

It is easily proved that prime gaps can be arbitrarily large by constructing the sequence of composites $(n+1)! + 2, (n+1)! + 3, \dots, (n+1)! + (n+1)$, which are divisible by $2, \dots, n+1$ ...
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3answers
90 views

Finding the possible Least Common Multiples of of numbers with Highest Common Factor 8

The Highest Common Factor of two numbers is 8. Which one of the following can never be their Least Common Multiple? The choices are as follow: A. 8 B. 12 C. 60 D. 72 The answer key states ...
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0answers
19 views

Asymptotics of $\sum_{\mathfrak{a}}\frac{n^{k-\epsilon}}{\mathfrak{N}\left(\mathfrak{a}\right)^{r\left(k-\epsilon\right)}}$

In this paper by Brian D. Sittinger, the following claim is made: For an algebraic number field $K$ with norm $\mathfrak{N}$, let $\epsilon=\left[K:\mathbb{Q}\right]^{-1}$. Then, taking the sum over ...
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2answers
30 views

all the squares in the multiplicative group $\mathbb{Z}_n^*$. [on hold]

I just want to know what this statement means: "all the squares in the multiplicative group $\mathbb{Z}_n^*$."
4
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2answers
48 views

Diophantine equation not solvable in $\mathbb{Q}$, but in $\mathcal{O}_p$

I'm trying to think of an example of a diophantine equation which can be solved in $ \mathcal{O}_p$ (meaning it can be solved $\mod p^k$ for all $ k $) for all prime $ p $'s, but not in $\mathbb{Q}$ ...
3
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1answer
41 views

Finitely Many Extensions of Fixed Degree of a Local Field

How does one show that there are only finitely many degree $n$ extensions of a local field? I understand how this follows from class field theory in the Abelian case but don't understand how to do the ...
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0answers
25 views

Hensel's lemma in $n $ variables

I'm trying to find a proof for the following formulation of Hensel's lemma: $$\text{Let } f \in \mathbb{Z}[x_1, \dots, x_n], a = (a_1, \dots, a_n) \text{ be such that (with } p \text{ prime)}$$ $$ ...
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2answers
273 views

Numbers that are divisible by the number of primes smaller than them

Let $\pi(n)$ denote the number of primes less than or equal to $n$ (a.k.a the prime-counting function). For certain values of $n$, the value of $\frac{n}{\pi(n)}$ is integer. Here are the first few ...
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1answer
23 views

Chevalley's theorem proof

I'm trying to prove Chevalley's theorem stating that $$ \text{If } f \in \mathbb{Z}[x_1, \dots, x_n] \text{ is a form of degree } r < n \text{,}$$ $$ \text{then there exists a nonzero solution of ...
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0answers
28 views

>Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$

Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$ I have to find $(m, n)$ such that ...
7
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0answers
72 views

Integer solutions to $\prod\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}x_i^2$

Given integers $x_1,\dots,x_n>1$. Let's assume WLOG that ${x_1}\leq\ldots\leq{x_n}$. I want to prove that the only integer solutions to any equation of this type are: $x_{1,2,3 ...
5
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1answer
73 views

Paul Erdős showed a simple estimate for $\pi(x) \ge \frac{1}{2}\log_2 x$; is it possible to tweak his argument to improve the estimate?

Paul Erdős gave a simple argument to show that $\pi(x) \ge \dfrac{1}{2}\log_2 x$. Is it possible to tweak the argument and get a better estimate? I am wondering how good an estimate for $\pi(x)$ can ...
2
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0answers
50 views

Any formula for the exact number of primes below a given bound?

Reading The music of the primes, the author relates that Riemann had figured out a formula giving exact number of primes up to a certain bound with no errors. Does such formula really exist? If ...
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3answers
34 views

Which residue classes modulo $p$ have exactly one square root?

Let $p>2$ be a prime. Which residue classes modulo $p$ have exactly one square root? Explain. I am having trouble understand the question. What does it mean for a residue classes to have exactly ...
2
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0answers
62 views

Prove that for $n\ge 6$ there is always a solution

We have the eation $\frac{1}{a_1^2} + \frac{1}{a_2^2}+...+\frac{1}{a_n^2}=1$. Prove that the equation has for $n\ge 6$ always natural solutions. Any $\frac{1}{x^2}$ can be displayed as sum of 4 ...
2
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1answer
43 views

Solve the eqation $p^8-p^4=n^5-n$

Solve the equation for primes $p$ and natural numbers $n$ $p^8-p^4=n^5-n$. For $p=2$ we get $n=3$, bjt for the next 5 prime numbers we get irational numbers. I cant prove (if its true) that there are ...
2
votes
2answers
33 views

Determine if the polynomial $P(x)=x^2-x+2 \mod p$ factors for the primes $p=5,7,11$, and $101$.

Determine if the polynomial $P(x)=x^2-x+2 \mod p$ factors for the primes $p=5,7,11$, and $101$. If it does factor for a particular prime provide a factorization. If not explain why. How would I be ...
3
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1answer
44 views

What proportion of the positive integers satisfy this number-theoretic inequality?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$, and let the abundancy index of $x$ be defined as $$I(x) = \frac{\sigma(x)}{x}.$$ My question is this: What proportion of the ...
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0answers
34 views

Sum of two squares - Number of steps in Fermat descent

If a prime $p$ can be written as the sum of two squares, then one can construct this representation via Fermat descent if we know an $x$ such that $x^2 \equiv -1 \mod p$. Is there a possibility to say ...
2
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4answers
66 views

Proving that $x^5 = x \pmod{10}$ for every integer $x$. [duplicate]

Show that $x^5 = x \pmod{10}$ for every integer $x$. How can I approach this? Should I use induction? I am stuck trying to get it in terms of $x+1$. Some feedback would be appreciated.
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1answer
70 views

Find all natural sequences $a_n=a_{a_{n-1}}+a_{a_{n+1}}$

Find all natural sequences for which holds $a_n=a_{a_{n-1}}+a_{a_{a+1}}$ a) for all natural numbers $n\ge 2$ b) for all natural numbers $n\ge 3$ I tried to do something with the caracteristic ...
0
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1answer
102 views

Finding the largest factorial of only three digits

I am using the following Python code to compute the above, but no results up to 16000!: ...
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0answers
26 views

Prove by induction on the binomial coefficient n choose m …

Prove by induction on $n$ that the binomial coefficient $\begin{pmatrix}n\\m\end{pmatrix}$ is the number of subsets of $I_{n}$ having size equal to $m$. The solution is as follows: So far it can be ...