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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0answers
22 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
5
votes
3answers
77 views

Can a square be in the form $2x + 1$, when $x$ is odd?

I was given this question, and I think I have solved it, but I'm not sure it is correct because this differs from how the answer is given. What is the most common way to solve this problem? Let's ...
2
votes
2answers
655 views

True or false identity?

I found the logo from The Eighth Congress of Romanian Mathematicians. I think this is the von Mangoldt summatory function and with a simple computation, using this definition, I obtained $83$. Am I ...
4
votes
0answers
47 views

A sequence avoiding 3-term power progressions

Rankin1 studied sequences of integers that avoid 3-term geometric progressions, $(a, a c, a c^2)$, e.g., $$\{1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, \ldots \} \;$$ So, $18$ is excluded ...
3
votes
1answer
34 views

Split Factorial of n

How can I split integers up to n into two groups such that the difference of the product of each group is as low as possible? Is there a way to optimize the selection for each group in order to ensure ...
2
votes
3answers
133 views

Markov-Hurwitz equation

Prove that the Markov-Hurwitz equation $x^2+y^2+z^2=dxyz$ is solvable in positive integers iff d= 1 or 3. Of course the reverse direction is easy, just set x=y=z=1, d=3. But I really have no idea ...
0
votes
0answers
33 views

Divisor sum function for integral values [on hold]

Let $d(n)$ be the number of divisors of a positive integer $n$. From the analytic theory it is known that the sum function of $d(n)$ up to non-integral $x$ is given by ...
2
votes
1answer
44 views

Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?

Maybe a strange (or stupid) question, but does $$\zeta(s)^2 \pm \zeta(1-s)^2$$ also have roots equal to the non-trivial zeros ($\rho$) ? At first sight you would expect so, however when I tried to ...
0
votes
1answer
57 views

Proof that $\sqrt{3} \notin \mathbb{Q}(\theta)$ where $\theta^4-2=0$. [closed]

This is a problem in Robert Ash's lecture notes in Algebraic Number Theory. I have to prove that $\sqrt{3} \notin K=\mathbb{Q}(\theta)$ where $\theta^4-2=0$, using the fact that ...
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2answers
51 views

Prove that to any three numbers positive integers [closed]

Prove that for any three positive integers, following equality holds $$\operatorname{lcm}(ab , bc , ca ) \cdot \gcd(a , b, c )=abc$$
1
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0answers
47 views
+50

Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
8
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5answers
174 views

The maximal size of between $\varphi(n)$ divided by $\lambda(n)$.

I want to find $$f(n) = \max\left\{\frac{\varphi(k)}{\lambda(k)} : 1 \leq k \leq n\right\}$$ In other words, I want to find the maximal value of $\frac{\varphi(k)}{\lambda(k)}$ when $k$ is ...
3
votes
1answer
101 views

A real number is rational $\iff$ its continued fraction expansion is finite.

I know that if this expansion is finite, then I can go to the lowest denominator in the whole fraction and turn it into a fraction and keep doing so until I get a fraction which means the number is ...
90
votes
1answer
3k views

All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$

Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have ...
3
votes
3answers
61 views

Is it necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? [closed]

As the title suggests, is it a necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? Like, I understand the dictionary definitions of necessary and ...
5
votes
1answer
80 views

Counting square free numbers co-prime to $m$

Counting square free numbers $\le N$ is a classical problem which can be solved using inclusion-exclusion problem or using Möbius function (http://oeis.org/A071172). I want to count square free ...
0
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3answers
44 views

How to find all primitive triples (a,b,c)? (Pythagorean Triples)

how to find all primitive triples when one value of (a,b,c) is given? For example in this case a = 45. What is the procedure to find the primitive triples ? Conditions for primitive triples are: ...
2
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2answers
59 views

what is the definition of numbers? [duplicate]

Well the question may seem obvious but I can't really find a proper answer to this. Mathematics all seem kind of difficult to understand so please help. I believe it is a quantity. Thanks in advance.i ...
0
votes
0answers
24 views

square classes of quadratic extensions of 2-adic fialds

I have a question about square classes of quadratic extensions of 2-adic fields. I appreciate anybody help me to understand. Why all elements of $1+\mathfrak{p}^5$ are square in ...
0
votes
0answers
42 views

Is the field of real algebraic numbers a complete field?

Let $\mathbb{R}_{alg}$ be the field of real algebraic numbers. Is there exist a metric $|\cdot|$ for which $(\mathbb{R}_{alg}, |\cdot|)$ is a complete field (i.e. any Cauchy sequence converges in ...
1
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0answers
30 views

sextic reciprocity and divisibility question

Regarding the question if $p|(2^{2(p-1)/6}+2^{(p-1)/6}+1) $ where $p$ is a prime of the form $7\mod 8 $ That is how far I got: $2^{(p-1)/6} \mod\ p\equiv x $ if the solution of $x^6\ mod\ ...
5
votes
0answers
182 views

One-to-one correspondance between zeta zeros and the prime powers?

I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2 $ in the ...
0
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1answer
38 views

Proving Bezout's Theorem

I need to prove Bezout's Theorem and the recommended method is using the induction on the number of steps before the Euclidean algorithm terminates for a given input pair.$~~~~~~$ I am having hard ...
8
votes
2answers
182 views

$\binom{2p-1}{p-1}\equiv 1\pmod{\! p^2}$ implies $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{\! p^2}$; where $p>3$ is a prime?

From $\binom{2p-1}{p-1}\equiv 1\pmod{\! p^2}$ how does one get $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{\! p^2},\,\forall a,b \in \mathbb N,\, a>b$; where $p>3$ is a prime ?
1
vote
1answer
31 views

Expected value of discrete functions. [closed]

I am doing some research in number theory(High school-so nothing advanced). During this I came across this post. I have not done much statistics. So could someone explain to me why if $\displaystyle ...
0
votes
0answers
14 views

Judge whether a function is $\mathcal{EF}$ or $\mathcal{PRF}$

All of the functions discussed below are total number-theoretic functions. Define two functions: $$ f:\mathbb{N}\to\mathbb{N},f(n)=\lfloor n\cdot e \rfloor \\ g:\mathbb{N}\to\mathbb{N},g(n)=\lfloor ...
0
votes
0answers
32 views

Pythagorean triangle with in-radius r: problems

If there is no odd prime divisor of $r$, prove that there is only one Pythagorean triangle with in-radius r. If $r=pq$, the product of two distinct primes, prove that there are four ...
0
votes
3answers
30 views

Problems on Pythagorean triangle

Show that there is one (no) Pythagorean triangle whose sides are in arithmetic (geometric) progression. The problem has two parts. There is one Pythagorean triangle whose sides are in arithmetic ...
3
votes
2answers
40 views

Find the last two digits of $33^{100}$

Find the last two digits of $33^{100}$ By Euler's theorem, since $\gcd(33, 100)=1$, then $33^{\phi(100)}\equiv 1 \pmod{100}$. But $\phi(100)=\phi(5^2\times2^2)=40.$ So $33^{40}\equiv 1 ...
3
votes
2answers
82 views

Existence of a specific prime

Reading some background on the Brocard-Ramanujan equation I found a proof which says the following. For a fixed natural number $A$ which is not a square it is always possible to find a prime $p$ such ...
1
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2answers
40 views

Show that $a^{16}-b^{16}$ is divisible by $133$ if $a$ and $b$ are both prime to $85$

Show that $a^{16}-b^{16}$ is divisible by $133$ if $a$ and $b$ are both prime to $85$ Since $(85, a)=1(17,5)$ and $(85, b)=(17,5)$ then $a^{16}-1\equiv (mod ~17)$, $a^{4}-1\equiv (mod~ 5)\implies ...
1
vote
4answers
109 views

Given primitive solution to $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$, show $a+b$ is a perfect square [duplicate]

If $a,b,c$ are positive integers and $\gcd(a,b,c)$ is $1$. Given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$ then prove that $a+b$ is a perfect square. I was trying to get something useful from ...
3
votes
4answers
51 views

Find remainder when $777^{777}$ is divided by $16$

Find remainder when $777^{777}$ is divided by $16$. $777=48\times 16+9$. Then $777\equiv 9 \pmod{16}$. Also by Fermat's theorem, $777^{16-1}\equiv 1 \pmod{16}$ i.e $777^{15}\equiv 1 \pmod{16}$. ...
-4
votes
1answer
15 views

simplification and number system [closed]

The sum of all digits except the unity that can be substituted at the place of k in order to be divisible by 8 in the number 23487k2 is?
0
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1answer
34 views

Find the missing digit in the number 23104*791

Find the missing digit in the number $23104*791$ if (i) it is divisible by $11$, (ii) it is divisible by $13$, (iii) it is divisible by $63$. (i) $23104*791=231 ...
4
votes
6answers
415 views

Elementary number theory - prerequisites

Since summer comes with a lot of spare time, I've decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really ...
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votes
0answers
33 views

find a method for twin primes and with Golbach conjecture [closed]

There are infinitely many twin primes. Two primes (p, q) are called twin primes if their difference is 2. Let be the number of primes p such that p<= x and p + 2 is also a prime. a sample ...
3
votes
7answers
78 views

What is the remainder when $6\times7^{32} + 7\times9^{45}$ is divided by $4$?

What is the remainder when $6\times7^{32} + 7\times9^{45}$ is divided by $4$ ? $7 \equiv 3 \pmod 4$ $7^2 \equiv 9 \pmod 4\equiv 1 \pmod 4$ $(7^2)^{16} \equiv 1^{16} \pmod 4$ i.e $7^{32} ...
0
votes
1answer
20 views

Show that $x_1x_2\cdots x_n (mod~ m)\equiv (x_1 (mod~m)\cdot x_2 (mod~m)\cdots x_n (mod~m))(mod~ m)$

Show that $x_1x_2\cdots x_n (mod~ m)\equiv (x_1 (mod~m)\cdot x_2 (mod~m)\cdots x_n (mod~m))(mod~ m)$ I know that $a\equiv b (mod ~ m)$, $c\equiv d (mod ~m)$ implies $ac\equiv bd (mod ~m)$ but how ...
0
votes
0answers
9 views

Where $a, b$ coprime, does $ax + b$ generate infinitely many 2-almost primes, infinitely many 3-almost primes, etc.?

I've seen various references to Dirichlet's theorem on arithmetic progressions claiming that where $a, b$ coprime, $ax + b$ not only generates infinitely many primes, but also infinitely many ...
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2answers
41 views

In how many ways I can write a number $n$ as sum of $4$ numbers?

The precise problem is in how many ways I can write a number $n$ as sum of $4$ numbers say $a,b,c,d$ where $a \leq b \leq c \leq d$. I know about Jacobi's $4$ square problem which is number of ways ...
4
votes
2answers
52 views

Use Wilson theorem to show that $63! + 1 \equiv 0 \mod ~ 71$

Use Wilson theorem to show that $63! + 1 \equiv 0 \mod ~ 71$. 71 is prime then Wilson theorem says that $(71-1)!+1=0 \mod ~ 71$ i.e $70!+1\equiv 0 \mod ~ 71$ then how to proceed further?
1
vote
1answer
36 views

Find out a process to generate pairs of distinct positive integer $m$, $n$ with $\phi(m) = \phi(n)$.

Find out a process to generate pairs of distinct positive integer $m$, $n$ with $\phi(m) = \phi(n)$. Attempt: The pairs $m=1, ~ n=2$; $m=3, ~n=4$ satisfy the problem. But I need a ...
48
votes
1answer
2k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
2
votes
1answer
41 views

Why is $x^{100} = 1 \mod 1000$ if $x < 1000$ and $\gcd (x,1000) = 1$?

Let $U(1000) =$ the multiplicative group of all integers less than and relative prime to $1000$. "Show that for every $x \in U(1000)$ it is true that $x^{100} = 1 \mod 1000$." Been thinking ...
11
votes
1answer
524 views

Deriving a trivial continued fraction for the exponential

Lately, I learned about the following continued fraction for the exponential function: $$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$ I thought it was ...
4
votes
0answers
25 views

Proving that there exist products of $a_k \equiv 1 \pmod {a_i}$ [closed]

Let $n>2$ be an integer. Prove that there exist numbers $a_1, a_2, \ldots ,a_n$ such that $$a_1a_2\cdots \widehat{a_i}\cdots a_n \equiv 1 \pmod{a_i}$$ for $i=1,2,3,\ldots,n$. Here ...
1
vote
3answers
59 views

Find solutions to $4x^2\equiv 1\pmod {29}$ [closed]

Find the solutions to the congruence: $$4x^2\equiv 1\pmod {29},\rm{ie},(2x)^2\equiv 1\pmod{29}$$
4
votes
0answers
57 views

Zeta zeros standard normal distribution

Below is a partially scaled plot of $\vartheta (\gamma_n) - \pi (n - 3/2) ,$ where $\gamma_n$ is the imaginary part of the $n$th zeta zero, and $\vartheta $ is the Riemann-Siegel theta function, for ...
1
vote
1answer
39 views

Non real complex in metric completions of $\mathbb Q$

Process of completion of $\mathbb Q$ using the absolute value $|x|$ does not touch to the non-real complex numbers which are added to $\mathbb Q$ via extensions fields. However completion of $\mathbb ...