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Questions tagged [number-theory]

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.

4
votes
0answers
38 views

Coloring the Natural Numbers

This problem appears on page $134$ of Peter Winkler's Mathematical Mind-Benders. Can you color the natural numbers $\{0,1,2,\dots\}$ with finitely many colors, in such a way that the sum $x+y$ ...
-1
votes
0answers
22 views

Show $\frac{\sum_{n=1}^{\infty}\frac{1}{\phi(n)^s}}{\zeta(s)}=\prod_{p}(1-1/p^s+1/(p-1)^s)$.

I want to show that the following equality and that the product is absolutely convergent and uniformally convergent on compact subsets of ${s:Re(s)>1}$. $$\frac{\sum_{n=1}^{\infty}\frac{1}{\phi(n)^...
0
votes
2answers
20 views

Can you propose a theorem from these congruences?

Find the least positive residues of a) $6! \ (\mod 7)$ b) $10! \ (\mod 7)$ c) $12 ! \ (\mod 13)$ d) $16! \ (\mod 17)$ e) Can you propose a theorem from above congruent? My ...
-3
votes
0answers
20 views

Non-trivial solutions of $Ax^2+By^2=Cs^2$ and $Ay^2+Bx^2=Ct^2$, where $A=p^2-q^2+2pq$, $B=p^2-q^2-2pq$, $C=p^2-q^2$ for integer $p$ and $q$

Given that, for integers $p$ and $q$, $$\begin{align} A &= p^2-q^2+2pq \\[2pt] B &= p^2-q^2-2pq \\[2pt] C &= p^2-q^2 \end{align}$$ it can readily be shown that the two conic sections $$\...
0
votes
0answers
36 views

Problem Similar to Ramanujan's Magic Square

This question is about this video about this "genius" kid Steve Harvey: A Mathematics Genius I was just wondering how was that possible. I felt that it was not just a training for that one particular ...
2
votes
0answers
35 views

Modular geometry: The parabolas of quadratic residues modulo $p$

[For using the available space better, I rotated the function graphs by 90 degrees.] For the quadratic function $f_1(x) = x^2$ (with $x \in \mathbb{R}$) there is only one parabola all integer points $...
0
votes
1answer
11 views

Existence of classical Dirichlet series from Lebesgue integrable inverse Mellin transform

Let $f(s)$ be meromorphic in $\mathbb{C}$. Let the following inverse Mellin transform be Lebesgue integrable for all real positive $x$ at some complex point $s$ with some real $c$: $\frac{1}{2\pi i} \...
0
votes
1answer
32 views

A question regarding primes and least common multiples

Let $p$ and $q$ be distinct primes. Show that the equation $\textrm{lcm}(a, b) = (p^2) q$ has fifteen solutions in positive integers $a, b$. Thanks for any help in advance as I don't know how to ...
0
votes
2answers
51 views

Multiple proofs of $\sum_{d|n}{\phi(d)}=n$ [duplicate]

I am looking for multiple proofs of that statement: here $\phi(n)$ denotes the Euler’s totient $$\sum_{d|n}{\phi(d)}=n$$ Here’s one: By unique factorisation theorem: $n=\prod_{k=1}^{m}{p_k^{\...
0
votes
1answer
52 views

$x^3=2y^3+4z^3$ in the set of integers [on hold]

If $x, y, z$ are integers solve: $$x^3=2y^3+4z^3$$
5
votes
2answers
125 views

Prove that the product of any two numbers between two consecutive squares is never a perfect square

In essence, I want to prove that the product of any two distinct elements in the set $\{n^2, n^2+1, ... , (n+1)^2-1\}$ is never a perfect square for a positive integers $n$. I have no idea on how to ...
0
votes
1answer
57 views

The proof that $\sqrt{q}$ is a rational number iff $q$ is a perfect square

I have a proof of that if $q\in \mathbb{Q}$ then $ \sqrt{q}$ is rational if and only if $q$ is a perfect square (it can be written in the form $q={p_1}^{a_1}...{p_n}^{a_n}$ where integers $a_j$, which ...
4
votes
1answer
30 views

Meaning of Hasse-Arf theorem

I am reading about the Hasse-Arf theorem in Serre's 'Local Fields' and I have a hard time understanding what exactly it means for the upper numbering to have jumps only at integers. It seems like a ...
2
votes
2answers
45 views

How to solve the equation $x^2-ny^2$ with $n>0$ and $p$ prime in integers?

I often see the equation $x^2+ny^2=p$ discussed where $n$ is a positive integer and $p$ is a prime. What about a negative value of $n$, particularly the equation $x^2-3y^2=p$?
0
votes
1answer
37 views

Higher bound for prime numbers [on hold]

Hello I'd like to know if this is true $p_{n+2}\leq 4(1+2+...+n)+4$ for all $n\in \mathbb{N}$. $p_{n+2}$ is the $(n+2)$-th prime number.
1
vote
1answer
23 views

Number of integers coprime to a given integer $q$ in some range $[x, x+y]$

I am asked to show that for $1 \leq x,y$ and an integer $q$, we have: $S(x,x+y,q) = |\{x < n \leq x + y \mid n \text{ is comprime to } q\}| = \frac{\phi(q)y}{q} + O(2^{\omega(q)})$, where: $\...
0
votes
3answers
37 views

Greatest common divisors [on hold]

If $\gcd(a,p)=1$ where $p$ is an odd prime then $\gcd(4a,p)=1.$ Any hint??
2
votes
2answers
62 views

Find at least $5$ integers $n$ such that $\varphi(n)=16$

Let $\varphi(n)$ denote Euler's totient function. Find all integers such that $\varphi(n)=16$. Answers given were $17,32,34,40,48.$ I am thinking a generalisation of this problem: is there a way ...
6
votes
2answers
84 views

Is there any elementary solution for this problem on colored interval?

The problem is as following. Assume $m,n$ are two coprime odd numbers, consider the interval $[0,mn]$. We cut the interval by $m,2m,\ldots,(n-1)m$ and $n, 2n,\ldots, (m-1)n$ into $m+n-1$ pieces of ...
6
votes
1answer
55 views

show $p\mid 2^n-n$ for infinitely many $n$ [duplicate]

Show that $p\mid 2^n-n$ for infinitely many $n$. $p$ is a prime and $n$ is an integer. I tried using Fermat's little theorem and got $2^p-p\equiv2\pmod p$ and $2^{p-1}-(p-1)\equiv2\pmod p$. So I can'...
2
votes
0answers
30 views

Constraints over $e$ for $x^n \equiv a (2^e) $ to be solvable

I'm reading "A classical intro to modern number theory" by Ireland & Rosen, in which Proposition 4.2.4 proof is left as an exercise. Let $n=2^lk $ with $k $ odd, assuming that $x^n\equiv a (2^e) $...
1
vote
1answer
23 views

Understanding this proof regarding quadratic residues

Let $p$ be an odd prime and let $Q_p$ denote the quadratic residues modulo $p$, $N_p$ the non-residues modulo $p$. Let $X$ be some subset of $p$. Then, $$ q X \equiv X mod (p) \hspace{2mm}\forall q \...
1
vote
1answer
79 views
+50

Mellin inverse of $\sum_{n=0}^{\infty}\frac{\Gamma(n+s)\zeta(n+s)\zeta(n+1+s)}{\zeta(s)n!}\left(-\omega\right)^{n}$

I am trying to compute the inverse Mellin transform of : $$\sum_{n=0}^{\infty}\frac{\Gamma(n+s)\zeta(n+s)\zeta(n+1+s)}{\zeta(s)n!}\left(-\omega\right)^{n}$$ w.r.t. the complex number $s$. $\omega$ ...
5
votes
1answer
74 views

Show that $n^2-1+n\sqrt{d}$ is the fundamental unit in $\mathbb{Z}[\sqrt{d}]$ for all $n\geq 3$

Let $n\in \mathbb{Z}$, $n\geq3$ and $d=n^2-2$. I want to show that $n^2-1+n\sqrt{d}$ is the fundamental unit in $\mathbb{Z}[\sqrt{d}]$. Substituting $n=3,4,5$ gives the elements $8+3\sqrt{7}$, $15+4\...
2
votes
2answers
36 views

Show that $n^2-1+n \sqrt{d}$ is always a unit in $\mathbb{Z}[\sqrt{d}]$

We let $n\in \mathbb{Z}$, $n>2$ and $d=n^2-2$. We want to show that $n^2-1+n\sqrt{d}$ is a unit of $\mathbb{Z}[\sqrt{d}]$. My initial idea was to consider the induced norm $N:R\to\mathbb{Z}$, ...
1
vote
0answers
58 views

“Exceptional” primes greater than 2

In many theorems, the prime $2$ has an exceptional characteristic. But where can other primes be exceptional? As a couple examples: 1) In the Fibonacci sequence $5$ is exceptional because $p=5$ ...
0
votes
0answers
31 views
+100

Shattering with sinusoids

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$, does there exist an ...
1
vote
1answer
63 views

Find all primes $p$ for which there are positive integers $x, y$ such that $p+1=2x^2$ and $p^2+1=2y^2$ [duplicate]

Find all primes $p$ for which there exist positive integers $x, y$ such that $p+1=2x^2$ and $p^2+1=2y^2$. I have tried coming up with an equation for $p$ or $p^2$ and this is what I've got $p=2x^2-1$...
3
votes
1answer
64 views

Show that $\Bbb Z[\alpha,4/\alpha]/(2)$ is a sum of three domains, where $\alpha^3+\alpha^2-2\alpha+8=0$

Let $\alpha$ be a root of $x^3+x^2-2x+8$. As part of another problem, I am trying to show that, in the ring $\Bbb Z[\alpha,4/\alpha]$, the ideal $(2)$ splits as a product of three distinct primes. ...
2
votes
1answer
42 views

Question related to derived formula for zeta-zero counting function

I've been attempting to derive a zeta-zero counting function based on the distributional or Fourier series representation of the second Chebyshev function $\psi(x)$ or its first-order derivative $\psi'...
2
votes
0answers
37 views

Density of integers $n$ with all prime factors of order $O(\log n)$?

For a rational integer $n \in \mathbb{Z}_{+}$, let $\mathfrak{p}(n)$ denote the set of (distinct) prime factors of $n$. Then for a positive constant $c$, let $$f(x) = \vert\{n\in\mathbb{Z}_{+}:\ n\...
0
votes
0answers
41 views

Permutations of quadratic residues modulo $p$

When bending the square $[0,n[\times[0,n[$ to a torus, the quadratic residues $k^2\ \%\ n$ — with $0 < k < n$ and $a\ \%\ b$ meaning $a$ modulo $b$ — lie on a parabola $P_1$ with ...
1
vote
2answers
73 views

Finding all rational points in $x^2+y^2=6$.

I want to find all rational points in the circle $x^2+y^2=6$. This would be easy if I could find one rational point in the circle, however, it's very hard to guess one in this case. However, I don't ...
1
vote
1answer
51 views

Since $\lim_{n\to\infty}\pi(n) = \frac{n}{ln(n)}$, can't this be used to prove Legendre’s conjecture?

Legendre's conjecture states that for all $n$, there is a prime number between $n^2$ and $(n+1)^2$. It has been proven that $\lim_{n\to\infty}\pi(n) = \frac{n}{\ln(n)}$. It can be proven that $\...
2
votes
1answer
64 views

Can somebody please answer this? [on hold]

Find all $(x,y)$ such that $x^2-1=y^3$
1
vote
4answers
98 views

a,b,c are three real numbers where $a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$. Now $abc$ = ? Here will the answer be a number? [duplicate]

I want to know whether it is possible to get a real number (not an algebraic expression) as the product of $a$, $b$ and $c$. I tried for a long time and this is what I got. $$3abc = a^2 b + b^2 c + ...
1
vote
0answers
21 views

Proof check: Showing gcd(a,b) has prime factorization with exponents taken to be minimum of those for a,b

Using the notation from the previous problem in the text, we've that $$a=t_1^{g_1}t_2^{g_2}\ldots t_v^{g_v} \quad \text { and } \quad b = t_1^{h_1}t_2^{h_2}\ldots t_v^{h_v},$$ for $t_1 < t_2 < \...
0
votes
3answers
340 views

Alternate definition of prime numbers

The prime numbers are usually defined to be positive integers with exactly two distinct divisors—one and the number itself. There are plenty of variations on this definition. Just out of sheer ...
1
vote
1answer
70 views

What is the value of $5 * 6$ in the following patttern?

Mr. Pascal built a computer for multiplying numbers and named it "Ramanujan". But Ramanujan multiplies $(3, 5), (2, 4), (3, 4)$ and $(4, 7)$and results are $17, 10, 14$ and $34$. If Ramanujan ...
2
votes
0answers
45 views

Convergence of $\sum\limits_p \frac{\chi(p)}{p}$ and the prime number theorem

Consider the sum $$\sum\limits_p (-1)^{\frac{p-1}{2}}\frac{1}{p}=-\frac{1}{3} + \frac{1}{5} - \frac{1}{7} - \frac{1}{11} + \frac{1}{13} + \frac{1}{17} - \cdots \tag{1}$$ of signed reciprocals of the ...
0
votes
1answer
51 views

General Quadratic Diophantine Equations of Three Variables

For a quadratic diophantine equation of two variables, $Ax^2+Bxy+Cy^2 =D$, it's not difficult to find the solutions as it is a generalized Pell equation. However, what happens when we incorporate more ...
1
vote
1answer
50 views

Prove the sum of the Mobius function over monic polynomials of degree $n$ is $0$ if $n > 1$

Let $\mu(m)$ be the Möbius function on monic polynomials in $\mathbb{F}_q[x]$ ($q$ is power of prime) where $\mu(m) = 0$ if $m$ is not square-free and $\mu(m) = (-1)^k$ if $m$ is square-free and can ...
2
votes
0answers
25 views

Gauss measure and continued fraction

For $x \in [0,1)$ then the continued fraction representation of $$x=0 + \cfrac{1}{a_1(x)+\cfrac{1}{a_2(x)+\cfrac{1}{a_3(x)+\cfrac{1}{\dots}}}}$$ can be written as $[0; a_1(x), a_2(x), a_3(x), \dotsc]$ ...
3
votes
1answer
68 views

Extending absolute values on local fields - what is the 'correct' normalization and the relation to the global theory?

I'm having a tough time figuring out the 'correct' normalization for extending absolute values of local fields. I'm also trying to piece together how this interacts with the global theory, so below is ...
0
votes
0answers
61 views

What a property does determine primitive root? [on hold]

We know that via a primitive root we can enumerate all elements of group by powering. The table elow shows the values of $a$ such that ${a}^{x}\equiv1\pmod7$ \begin{array}{c|c}a{\LARGE{\setminus}} x&...
2
votes
2answers
86 views

Two Subsets of Squares of Reciprocals of Primes with Equal Sums

Let $$A=\{\frac{1}{2^2},\frac{1}{3^2},\frac{1}{5^2},...\}$$ be the set of squares of the reciprocals of prime numbers. We have $$\sum_{x\in A}x < \infty$$ Do there exist $B \subset A$, $C \subset ...
-1
votes
2answers
41 views

If $p$ is prime and $\gcd(a,p)=1$ then $p\nmid a$. [on hold]

Prove that If $p$ is prime and $\gcd(a,p)=1$ then $p\nmid a$.Any idea?
-3
votes
1answer
39 views

Expressing the abc conjecture in symbols [on hold]

Is it possible to express the abc conjecture explicitly in mathematical symbols, without using any words?
4
votes
0answers
72 views

Balls and Boxes

Three boxes contain balls. Each box is large enough to contain all balls. We call $\bf{target box}$, a box that receive balls from one of the others boxes. We allow only one process: moving $n$ balls ...
0
votes
0answers
34 views

Book about Prime numbers [duplicate]

Is there any book that tells theorems about prime numbers with proofs. I know undergraduate calculus. I need a book that explains everything in detail.