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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2
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1answer
46 views

Number of solutions to this nice equation $\varphi(n)+\tau(n^2)=n$

How many natural numbers $n$ satisfy the equation$$\varphi(n)+\tau(n^2)=n$$where $\varphi$ is the Euler's totient function and $\tau$ is the divisor function i.e. number of divisors of an integer. I ...
2
votes
0answers
18 views

On equivalence of RSA and factoring

Suppose we are given a number "$A$" which is multiple of $\phi(n)$. One can assume factorization to be hard. So you cannot find exact value of $\phi(n)$ from $A$. Clearly using this we can crack ...
1
vote
1answer
8 views

How to inverse this equation which can give the devisors values of numbers

The roots of the following equation $f(x) = sin^2(\pi*x) + sin^2(\pi*n/x)$ are the positive and negative devisor of ($n$) For example, if we set ($n=7$) then ($x$) will have 4 real values {-7,-1,1,...
1
vote
1answer
69 views

Conjecture about primes and the faculty

Conjecture: Given a prime $p>5$ there exist a prime $q<p$ such that $kp+q=m!$, for some $k,m\in\mathbb Z_+$ where $m>2$. I want help to prove the conjecture or to find a counter-...
1
vote
1answer
119 views
+50

Why it is impossible for primitive Pythagoras triplets in integers to be all as powerful numbers?

I had seen an elementary proof for Fermat's last theorem at Quora. I had checked all the steps (around one page only),where I couldn't catch any error, but I was confused about the last step only ...
1
vote
1answer
23 views

Why is it not known if Mill's constant is rational or irrational?

The following text appears in the Mill's constant definition at the Wikipedia: There is no closed-form formula known for Mills' constant, and it is not even known whether this number is rational (...
4
votes
0answers
54 views
+100

How to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]?

$\forall \alpha\in [0,1]\setminus\mathbb{Q}$, how to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]? $\{x\}$ is the fractional part of x. Any hint would be appreciated!
2
votes
1answer
42 views

Proving a particular infinite continued-fraction identity

By iterating the basic relation $$ \forall z \in \mathbb{C} \setminus \{ -1 \}: \quad z = 1 + \frac{z^{2} - 1}{z + 1}, $$ one obtains the following finite continued-fraction identities: \begin{alignat}...
1
vote
2answers
62 views

Finding all possible values of a Function

Let a function be defined as $f:N\to N$ and $x-f(x)= 19\left[\dfrac{x}{19}\right] - 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$ and $1900<f(1990)<2000$. Find all values of $f(1990)$. $...
2
votes
3answers
123 views

A certain unique rotation matrix

One can find that the matrix $A=\begin{bmatrix} -\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & -\dfrac{1}{3} & \dfrac{2}{3} \\ \...
0
votes
3answers
46 views

Counting non-negative integral solutions

I'm reading this passage and wondering why Number of ways in which k identical balls can be distributed into n distinct boxes = $$\binom {k+n-1}{n-1}$$ could someone explain it to me please?
3
votes
3answers
31 views

Pattern on last digits of numbers to a certain power

There are 4 one-digit numbers which when squared have a last digit equal to the first number. They are 0,1,5 and 6. There are 2 two-digit numbers which when squared have their last two digits equal ...
3
votes
1answer
32 views

Show that every square n is congruent to $ 0$ or $1 \pmod{8}$

If $n$ is odd then $n$ is congruent to $1 \pmod{8}$, but if $n$ is even the we have to do it by cases. Let $n=2k$ so $n^2=4k^2$. When $k$ is even, then it's congruent to $0 \pmod{8}$, but when $k$...
2
votes
1answer
52 views

Olympic Problem about Theory of numbers.

Let $Y=\{1,2,\ldots, 2014\} \subset \mathbb{N}$. Find the maximal subset $A\subset Y$ such that, $$\forall x\in A,\quad x\not\mid\sum_{y\in A\setminus\{x\}} y.$$ Example, $A'=\{2,4,6,\ldots,2014\}\...
0
votes
2answers
25 views

Show that if $a\equiv b(modm)$, then $gcd(a,m)=gcd(b,m)$ [duplicate]

I still don't have a clear approach , but this is what I see. $m|b$ and $m|a$ or $m\nmid b$ and $m\nmid a$. I may think that the way is showing $gcd(a,m)\leq gcd(b,m)$ and $gcd(a,m)\geq gcd(b,m)$
13
votes
6answers
294 views

Is $ \sin: \mathbb{N} \to \mathbb{R}$ injective?

I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. ...
0
votes
0answers
26 views

On Oblath's Problem [on hold]

I am trying to read the paper On Oblath's Problem, and I'm have difficulty understanding the main theorem. I can read the theorem but I don't understand it. May someone help me to make this theorem as ...
12
votes
1answer
444 views

If $p$, $q$ are naturals, solve $p^3-q^5=(p+q)^2$.

In If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$., the author asks to solve the equation $p^3-q^5=(p+q)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this "...
1
vote
0answers
20 views

Lucas recurrence relation

Lucas recurrence relation is defined as: $V_n = PV_{n-1} – QV_{n-2}$; for $V_0 = 2; V_1 = P$ Here $P$ is positive integer and $Q = {-1, 1}$ or may be $+1$ or $-1$ A Fibonacci Pseudo prime with ...
1
vote
0answers
21 views

Devise a test for divisibility of an integer by 11, in terms of properties of its digits [duplicate]

Using the fact that $10 \equiv-1 \pmod{11}$, devise a test for divisibility of an integer by $11$, in terms of properties of its digits. Approach: Let the number with its digits $a_0\cdots a_n$ be ...
0
votes
0answers
9 views

Proof with $\sum_{i=1}^k \log_{b}\left(1-10^{-(m-i+1)}\right)$

Let $b>2$ and $1 \leq k \leq m$ where $a,b,k,m$ are integers. Prove that $$0.\overline{2}_b+\sum_{i=1}^k \log_{b}\left(1-b^{-(m-i+1)}\right)<\log_{b}{2}.$$ According to this question we know ...
0
votes
1answer
36 views

How Changing the order of integration(Elementary proof of the prime number theorem)?

I'm studying the exchange of integration order, I need help, any hint? For every real number $\rho \geq 0$, write $V(\rho)=e^{-\rho}R(e^{\rho})=e^{-\rho}\psi(e^{\rho})-1$ where $\psi(x)$ is the ...
1
vote
2answers
27 views

Prove that $\log_{10}(1-x^{-m}) \geq -2 \cdot x^{-m}$ for $x>2^{1/m}$.

Prove that $\log_{10}(1-x^{-m}) \geq -2 \cdot x^{-m}$ where $m,x > 2^{1/m}$. I was thinking about the analogous property for the base $e$ logarithm for which it is true that $\log(1-x) >-2x$ ...
0
votes
1answer
25 views

Perfect square from a multiple of factorials

I have a problem with this question John writes the number 1!, 2!, 3!, ... , 199!, 200! on a whiteboard. John then erases one of the numbers. John then multiplied the remaining 199 numbers. He found ...
0
votes
1answer
22 views

Find the least $m$ such that $0.\overline{2}_b<\log_{b}{2}$

Find the least $m$ such that $0.\overline{2}_b<\log_{b}{2}$ for all $b>m$. I didn't see a nice analytic way of solving this, so I was wondering if there was an easier way of solving it.
1
vote
2answers
32 views

Proof about congruence

Let $$f(x)=a_0x^n+a_1x^{n-1}+...+a_n$$ where $a_0,...,a_n$ are integers.Show that if d consecutive values of (i.e, values for consecutive integers) are all divisible by the integer d, then $d|f(x)$ ...
1
vote
0answers
61 views

Prove, that the product of $3$, and $4$ following natural numbers can never be a number with the form of $x^k$

Prove, that the product of $3$, and $4$ following natural numbers can never be a number with the form of $x^k$ , where $x$ and $k$ are natural numbers, and $k>1$ (for example $9$ has this form, ...
420
votes
33answers
48k views

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
0
votes
0answers
19 views

Representation of Frey's curve.

I read that Frey's curve is a semi-stable elliptic curve. What doe this mean ? I can find 2 dimensional representations of y^2 = x^3 + ax + b in Wikipedia. What does y^2 = x(x-a)(x+b) look like if a ...
1
vote
1answer
26 views

Remainder modulo matrices.

Is it possible to have a consistent definition of $A\bmod_{left} B$ that respects matrix multiplication from left where $A$ and $B$ are two square matrices with non-negative entries? Is there a ...
8
votes
0answers
153 views
+50

Solving (n+1)(n+2)…(n+k)−k = x^2

Let $n$ and $k$ be positive integers. Need to find all pairs of $(n,k)$ such that $$(n+1)(n+2) \cdots (n+k)−k = x^2,$$ where $x^2$ is a perfect square.
3
votes
1answer
69 views

What is the first square in the sequence $4729494n+1$?

Today I found a strange phenomenon that I want to ask about. If $$f(n)=4729494n+1,$$ is square, where $n$ is positive integers. Then I found $n=4729492$, because $$f(4729492)=4729493^2$$ In fact,...
2
votes
0answers
30 views

Proof of the Ribet's theorem

My question is very simple : My goal is to read a proof the proof of the epsilon conjecture proven by Ken Ribet (1986) which is an ingredient of the proof of the Fermat Last Theorem (I want the ...
2
votes
1answer
31 views

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$ i.e. $[-1] \ne [2^k] , \forall k >0$ in $\mathbb Z_n$ ? Is there any ( at least sufficient ) ...
1
vote
1answer
39 views

Composite integers $n$ such that ALL factors of $2^n-1$ $\equiv$ $1$ $\pmod n$?

Are there any composite integers $n$ such that ALL factors of $2^n-1$ $\equiv$ $1$$\pmod n$. (In other words, every prime factor dividing $2^n-1$ has the form $2kn+1$) It seems unlikely that there ...
3
votes
0answers
71 views

Continued fraction $1 + \frac 2{3 + \frac 4 {5 + \cdots}} = \frac 1 {\sqrt{e} - 1}$?

I saw this link (written in Japanese) and found an interesting problem: Calculate $1 + \frac 2{3 + \frac 4 {5 + \cdots}}$. The link provides the answer ($\frac 1 {\sqrt e - 1}$) and a hint that one ...
-1
votes
0answers
32 views

The order of the group $U(n)$ is even for $n\gt2$ [on hold]

Use the corollary to Lagrange's theorem that the order of an element in a group $G$ divides the order of the group $G$ to prove that the order of $U\left ( n \right )$ is even when $n\gt2.$ I ...
1
vote
0answers
61 views

Are complex numbers complete in every way?

I was told many times a story. Indeed a fascinating one to me as a student learning mathematics. First there were natural numbers. People started adding things and finding solutions to finding the ...
1
vote
0answers
133 views

solve $x^y-y^x=xy^2-19,$ $x,y\in\mathbb{Z}$

I have been struggling to solve this exercise but with no result: $$x^y-y^x=xy^2-19,$$ $x,y\in{\mathbb Z}$ I have started to think it has no solutions at all. I have no idea how to solve it so I was ...
7
votes
2answers
139 views

Is the equality $1^2+\cdots + 24^2 = 70^2$ just a coincidence?

I have read a question (written in Korean) that the equality $$1^2+2^2+\cdots + 24^2 = 70^2$$ is just a coincidence or not. It is a related to the integral points of the following elliptic curve (?): $...
1
vote
1answer
30 views

Decimal representation begins with $N$

Let $a \geq 2$ be a positive integer such that $\log_{10}a$ is irrational. Then for any positive integer $N$ there exist infinitely many powers $a^n$ of $a$ whose decimal representation begins with $N$...
1
vote
2answers
29 views

$7^n$ contains a block of consecutive zeroes

Prove that there is a positive integer $n$ such that the decimal representation of $7^n$ contains a block of at least $m$ consecutive zeros, where $m$ is any given positive integer. I will prove it ...
2
votes
3answers
47 views

Prove that $\sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$ [on hold]

Prove that $\displaystyle \sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$. I tried using the partial fraction decomposition $a_j = \frac{1}{2j} - \frac{1}{j+1} + \frac{...
1
vote
0answers
17 views

Sum over square divisors is multiplicative proof verification

I would like someone to verify my proof of the following claim, which I have been using to solve some problems about proving series identities in Ch. 11 of Apostol's analytic number theory text. Let $...
0
votes
1answer
52 views

Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find square root for Number Field Sieve(reference here page 4 first line). It says that for this method the degree of extension K/Q must be odd so that Norm(-x)=-...
1
vote
3answers
78 views

What is the meaning of $\Bbb{Q}[x]/f(x)$?

I am very confused with the meaning of $\Bbb{Q}(x)/f(x)$. Does it mean the set of all polynomials modulo $f(x)$? If it does then how can we say that $\Bbb{R}[x]/(x^2+1)$ is isomorphic to set of ...