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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Positive integer solutions to $x^2+y^2+x+y+1=xyz$

The question asks for positive integer solutions to $x^2+y^2+x+y+1=xyz$ . We at first note that $x|y^2+y+1$. Now,let there exist positive integers $x,y$ that satisfy the given equation.Then ...
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11 views

Extension of an arithmetic function like Gamma function

As you know Gamma function is an extension of the factorial function. Is there any other function which is the extension of an arithmetic function? For example extension of the Euler's totient ...
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2answers
28 views

integer solutions to $x^2+y^2+z^2+t^2 = w^2$

Is there a way to find all integer primitive solutions to the equation $x^2+y^2+z^2+t^2 = w^2$? i.e., is there a parametrization which covers all the possible solutions?
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1answer
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$n$ is some natural number. Let $x$ be the integer part of $\sqrt n$ and $y$ be the decimal part. If $x^2 - y^2 = 1+4y$ what is $y^x$?

$n$ is some natural number. Let $x$ be the integer part of $\sqrt n$ and $y$ be the decimal part. If $x^2 - y^2 = 1+4y$ what is $y^x$? This is some high school problem but I can't solve it. Any help? ...
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12 views

Group of numbers with common euler's totient function result

I was asked to find the group of integers, which share the result of euler's function of 84. To be clear: which numbers, when applying eulers function on them, result 84. By calculating I found that ...
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0answers
8 views

Rewriting $\tau(p)\Delta(\tau)$ when $p$ is prime

$p$ is a prime, and $\tau$ is Ramanujan's tau function: $$p^{11}\Delta(p\tau)+\frac{1}{p}\sum_{k=0}^{p-1}\Delta\bigg(\frac{\tau + ...
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3answers
34 views

If $\gcd(a,b)=\gcd(a,c)=1$ then $\gcd(a,bc)=1$

I need help in this. If $a,b,c \in \mathbb{Z}$ and $\gcd(a,b)=\gcd(a,c)=1$ then $\gcd(a,bc)=1$.
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2answers
66 views

Is it possible to reach 50k in the game 2048 with only having a highest tile of 512?

Is it possible to reach a score of 50,000 in the game 2048 with only having a highest tile of 512(Only one 512 was present) and one 256 tile and other small numbered tiles such as 2,4,8,16,...,64. ...
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1answer
62 views

Derivative of Riemann zeta, is this inequality true?

Is the following inequality true? $$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$ This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches ...
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1answer
23 views

If $\frac1\alpha+\frac1\beta=1$, irrational, then $\{\lfloor n\alpha\rfloor:n\in\Bbb N\}\uplus\{\lfloor n\beta\rfloor:n\in\Bbb N\}=\Bbb N$

Let $\alpha,\beta\in\Bbb R\setminus\Bbb Q$ such that $\frac1\alpha+\frac1\beta=1$, and define $S(x)=\{\lfloor nx\rfloor:n\in\Bbb N\}$. (Note that my convention takes $0\notin\Bbb N$.) The claim is ...
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1answer
125 views

Congruences and prime powers

I have just a small question that probably is not hard to answer, but I could not find and elegant solution to this question. Let $p$ and $q$ be prime numbers. $$5^q\equiv 2^q \pmod p$$ $$5^p\equiv ...
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1answer
36 views

If $d|n$, then $\phi(d)|\phi(n)$

Where $\phi(n)$ denotes Euler's Totient Function. My proof follows, I was hoping someone could verify it, and give critique. Let $d,n\in\mathbb{Z}^+$ so that $d|n$. By the Fundamental Theorem of ...
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0answers
10 views

Question regarding the sum of the reciprocal of the values in Sylvester's sequence

The unit fractions formed by the reciprocals of the values in Sylvester's sequence generate an infinite series: $\sum_{i=0}^{\infty} \frac1{s_i} = \frac12 + \frac13 + \frac17 + \frac1{43} + ...
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24 views

Determing sequence from its Dirichlet series

Suppose I know the Dirchlet series $$\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \frac{\zeta(s)}{\zeta(3s)},$$ where $\zeta(s)$ is the usual Riemann zeta function. My question is - is there a way to ...
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1answer
22 views

For which n does the inequality $2 \uparrow^{n+1}n > 3\uparrow^n 3 +2$ hold?

For which n does the following inequality hold ? $$2 \uparrow^{n+1}n > 3\uparrow^n 3 + 2$$ where $\uparrow$ stands for knuth's up-arrow notation. I need this inequality to prove that ...
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2answers
35 views

Is there any use of this mu function?

Let $\nu(\lfloor x\rfloor)$ be the function that gives $0$ if $\sqrt{⌊x⌋}$ isn't irrational, $1$ if $\sqrt{⌊x⌋}$ is an irrational number, and $-1$ if $\sqrt{⌊x⌋}$ is a rational number other than $1$ ...
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1answer
45 views

Getting multiple of 11

Take any $2$ digit number having different digits. Now add the bigger digit in that number. By continuing this process, you will get a multiple of $11$ i.e. both of the digits will be equal of a ...
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2answers
231 views

Find last n for which 2^n has a 0.

Find last number $n$ for which $2^n$ has a zero. For example $2^{10}=1024$ has a zero for which last number zero will be there. (It is possible that there doesn't exist such limits to $n$ but what is ...
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1answer
34 views

$\Bbb Z[i\alpha]$ UFD's

I know that $\Bbb Z[i]$ and $\Bbb Z[\sqrt{-2}]$ are Unique Factorization Domains, and that $\Bbb Z[\sqrt{-6}]$ is not. I have two questions. I know that they may be difficult questions, so I only ask ...
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1answer
24 views

Asymptotic result about analytic number theory

I don't know if there is any done work done about ehis matter, and I don't have access to research news. I'm interested in this question (I haen't tried to answer it myself, but it seems very ...
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0answers
42 views

Is there a known method for finding extremely huge squarefree numbers?

People often compete to beat the record for largest known prime (it is currently $2^{57,885,161}-1$). There are also big money prizes for finding explicit prime numbers exceeding specific magnitudes. ...
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1answer
24 views

Exponential generating function for number of 10 length sequences built from the alphabet, with some restrictions

I've got the following homework question. If anybody could possibly point me in the right direction, that would be great: Suppose X is a sequence with 10 terms built from 26 letters {a, b, c, ..., ...
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2answers
38 views

A quadratic equation over $\mathbb{Q}_p$

Suppose we have the equation $x^2+x+1$ over the field $\mathbb{Q}_p$. is it possible to determine for what primes $p$ the equation has solutions? I tried to see whether this is related to what $p$ is ...
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99 views

Solving $\phi(n)=84$

Ok, I really need some help understanding this because either my brain isn't working at the moment or I'm breaking math and I have a striking suspicion that one of those is more likely. Anyways, ...
2
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1answer
21 views

Properties of divisibility

I would like to know if it's true if $a|b$ and $c|d$, then $ac|bd$. I prove in this way: if $a|b$ and $c|d$, then there are $k_1$ and $k_2$, such that $b=k_1a$ and $d=k_2c$, thus $bd=k_1k_2ac$ and ...
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2answers
154 views

Discrete math: probability of picking certain hands with a preset condition

In 5-card draw poker, a player receives an initial hand of 5 cards, and is then allowed to replace up to three of her cards with the remaining cards in the deck. (b) Suppose that, among the initial 5 ...
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1answer
34 views

Rational vs irrational

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
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1answer
21 views

Inverse euler totient procedure

Given that if $n = p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ we know that $$\phi(n) = p_1^{\alpha_1 -1}(p_1 - 1) \cdots p_r^{\alpha_r -1}(p_r-1). \quad (1)$$ So, if $\phi(n)$ was given, the method of ...
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1answer
65 views

Dense sequence in $[0,1]$

There is the theorem proved by Liouville which states that if $x$ is irrational then there are infinitely many fractions $\frac{p}{q}$ such that $|x-\frac{p}{q}|<\frac{1}{q^2}$, i.e. ...
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1answer
47 views

Finding all primes $(p,q)$ for perfect squares.

Find all prime pairs $(p,q)$ such that $2p-1, 2q-1, 2pq-1$ are all perfect squares. Source: St.Petersburg Olympiad 2011 I have only found the pair $(5,5)$ so I am thinking that maybe a modulo $5$ ...
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0answers
29 views

What are some algorithms that can be used to test if a number is transcendental or not?

Well according to the definition of transcendental numbers I find that its any number that doesn't have any polynomial equation of any degree with integer coefficients summing up to 0. So ...
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3answers
28 views

Modular Arithmetic

I am doing some exam preparation and can't figure out how to do the following question. It seems to be a regular question and was wondering if anyone who could tell me an approach to this style of ...
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2answers
22 views

How to measure monotonicity of a list of values

I need to compare monotonicity of lists of values. I have $S=(n_1,n_2,...n_n)$, I need a function $\mathrm f(S)$ to return the monotonicity of the S. $S_1=[1,2,4,4,8]$ $S_2=[8,4,4,2,1]$ ...
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26 views

Given n , what is the sum of all gcd integers upto n with n? [duplicate]

Given an integer n, I want to find S = gcd(1,n) + gcd(2,n) + gcd(3,n) + ....gcd(n,n). Now , there are I have firgured that the number should be something like S = φ(n) + x. Now I can't draw a ...
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1answer
21 views

Divisiblity test for a random number n

If a number n has 60 divisors and 7n has 80 divisors, what is the greatest power of 7 that divides n. a)1 b)2 C)3 d)4
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1answer
22 views

Number Theory Prime Factor Problem

There is an integer N that has 12 factors, including 1 and itself, but only 3 of them are prime factors. The sum of these three prime factors is 20. What is the smallest possible value for N?
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67 views

How find this $5xy\sqrt{(x^2+y^2)^3}$ can write the sum of Four 5-th powers of positive integers.

Find all positive integer $x,y$ such $$5xy\sqrt{(x^2+y^2)^3}$$ can write the sum of Four 5-th powers of positive integers.In other words: there exst $a,b,c,d\in N^{+}$ such ...
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0answers
27 views

Legendre's Conjecture

I have read and heard conflicting reports about whether or not Legendre's conjecture has been proven. Refresh: Legendre believed that there will always be at least one prime between (n)^2 and (n+1)^2. ...
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2answers
63 views

Last Two Digits Problem

I'm trying to find the last two digits of ${2012}^{2012}$. I know you can use (mod 100) to find them, but I'm not quite sure how to apply this. Can someone please explain it?
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1answer
15 views

Question regarding solving a modulo equality

Two Equations: ab % c = d (ci + d) % c = d, i $\in \mathbb N$ I want to solve for b given the above two equations with a, c, and d known. ab = ci + d b = (ci + d) / a i = (k + an), n $\in ...
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2answers
43 views

If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$?

If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$? I am shamelessly asking how to solve the problem? I have no idea how to start and solve. Please help.
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4answers
49 views

Count the divisors of n with particular property

Take $n = \prod_{i=1}^r {p_i}^{\alpha_i}$, where each $p_i$ is a prime and $\alpha_i\geq 1$. How many divisors of $n$, not equal to $n$, contain at least one $p_i$ with the corresponding multiplicity ...
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2answers
43 views

Proof about Number Fields

It is a known result that if $\alpha$ is an algebraic integer in a number field $K$, i.e. $\alpha \in \mathcal{O}_K$, then its trace and norm are integers. I am looking over a proof of this, which ...
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1answer
37 views

How to know if the mth root of n is an integer?

If n can be represented in binary as a x bit integer, is there any algorithm such that we can determine if the mth root of n is an integer in time polynomial of x ?
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1answer
26 views

How prove this $\sum_{t|n}(d(t))^3=\left(\sum_{t|n}d(t)\right)^2$

show that $$\sum_{t|n}(d(t))^3=\left(\sum_{t|n}d(t)\right)^2$$ where $d(n)$ is the number of positive divisors of $n$. see this have simaler $$1^3+2^3+\cdots+n^3=\left(1+2+\cdots+n\right)^2$$ maybe ...
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2answers
27 views

How prove or disprove $\gcd(lcm[a_{1},a_{2},\cdots,a_{n}],a_{n+1})=\cdots$

let $a_{i},i=1,2,\cdots,n,n+1$ be positive integer numbers,prove or disprove $$\gcd([a_{1},a_{2},\cdots,a_{n}],a_{n+1})=[\gcd(a_{1},a_{n+1}),\gcd(a_{2},a_{n+1}),\cdots,\gcd(a_{n},a_{n+1})]$$ ...
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2answers
16 views

On extracting primes from coprimes

Proof or disprove the following statement - There exists infinitely many $a$ and $b$ which are pair of co-prime integers , either $ab+1$ or $ab-1$ is prime. Motivation- Looking at some twin prime ...
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3answers
32 views

Number Theory Remainder Question

I'm trying to find the answer to the following: What is the remainder when 9^2012 is divided by 11? Apparently, you're supposed to use Fermat's Little Theorem, but I'm not sure how to use it to solve ...
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1answer
19 views

Equation with a sum for the prime-counting function involving the Mobius function

I have come across the statement that $$ \sum_{n\leq x}\sum_{d\mid(n,P_z)}\mu(d) = \sum_{d\mid P_z}\mu(d) \left[\frac{x}{d}\right], $$ where $P_z=\prod_{p\leq z}p$ where $p$ is prime, $\mu(d)$ is the ...
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0answers
57 views
+50

Applications of generating functions to number theory

I am familiar (at least at a cursory level) with the extensive role generating functions play in the theory of partitions. What are some other prominent applications of generating functions to number ...