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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Coprimality of $2^n + 3^n$ and $5^n + 7^n$

Prove or disprove that for all positive integers $n$, $2^n +3^n$ and $5^n + 7^n$ are always coprime. This is not a homework problem, it is merely a problem I set myself after doing some number theory ...
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1answer
47 views

How to establish $\sum_{d|n}d\phi(d)$

I am focusing on #5(b). I do not understand how they go from what I have to the answer. Those are r's at the end.
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4answers
55 views

Problem modulo $p$.

Let $p$ be a odd prime, prove that $1^p+2^p+...+(p-1)^p \equiv 0 \mod p$ I'm not sure how to do this, thanks.
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1answer
46 views

Prove that for $n\ge 2$, the n-th Lucas number is equal to $[a^n+1/2]$

Prove that for n greater than or equal to 2 the n-th Lucas number is equal to $[a^n+1/2]$. The brackets are the greatest integer function, $a = \frac{1+\sqrt5}{2}$. I get every kind of proof we ...
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2answers
814 views

Show that every nonzero integer has balanced ternary expansion?

show that every nonzero integer can be uniquely represented in the form $e_k3^k + e_{k-1}3^{k-1}+ … + e_13+e_0$ where $e_j= -1, 0, 1$ for $j = 0,1,2,…k$ and $e_k \neq 0$
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1answer
48 views

Details about Generalized Convolution (Number Theory - Apostol)

In "Introduction to analytic Number Theory" by Apostol there is chapter about generalized convolution. Let F denote a real or complex-valued function defined on the positive real axis such that ...
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0answers
63 views

The zeros of $2\,\xi(s)-1$. Is there anything known about the curves they lie on?

Take the well known integral: $$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}-1} + {x}^{\frac{-s}{2}-\frac12}\right)\,\psi(x)\, ...
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3answers
62 views

Triangular numbers for numbers.

Interestingly for triangular numbers: $X(X+1)+Y(Y+1)=Z(Z+1)+a$ $a$ - this number is determined by the condition of the problem. Are all numbers equation has a solution? And what kind of formula in ...
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1answer
79 views

Generalized Pythagorean triples construction.

All primitive Pythagorean triples $(a, b, c) : \{ a^2 + b^2 = c^2 \} \wedge \{ a \equiv 0 \pmod{2} \}$ can be expressed in the form:$$\{ a = 2pq, b = p^2 - q^2, c = p^2 + q^2 \}$$ for positive ...
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1answer
85 views

If Cramér's is proved?

Harald Cramér proved that under this assumption that the Riemann hypothesis is true., the gap $g_n$ satisfies $$g_n = O(\sqrt{p_n} \ln p_n) ,$$ using the big O notation. Later, he conjectured that ...
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0answers
59 views

Calculate the number of times that $2$ appears in the prime-factorization of a given number?

Given a positive odd integer $K$ with a prime-factorization of $\prod\limits_{i=1}^{m}{P_i}^{N_i}$, is there a mathematical method for calculating the number of times that $2$ appears in the ...
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2answers
222 views

Is (the proof of) Fermat's last theorem completely, utterly, totally accepted like $3+4=7$?

If a mathematician would/does make use of Fermat's last theorem in a proof in a publication, would s/he still make use of some kind of caveat, like: "assuming Fermat's last theorem is true" or ...
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1answer
65 views

Ramification and roots of unity in complete discrete valuation rings.

Let $\mathcal{O}$ be a complete discrete valuation ring with algebraically closed residue field $k$ of characteristic $p>0$. Let $\pi\in \mathcal{O}$ generate the maximal ideal and suppose ...
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3answers
374 views

Testing If a Three/Four Digit Number is Prime or Not

Thank you for providing such great help. Thanks to math.stack site. I would like to know a good method to test any three/four digit number prime or not? I don't want to go any C or Java or any ...
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1answer
29 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
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3answers
51 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
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1answer
87 views

$n^2$ divides $\phi(a^n-1)$ whenever $n$ is even and $a>2$

My problem is to show that $n^2$ divides $\phi(a^n-1)$ whenever $n$ is even and $a>2$. I have thought a solution but it is quite long and tedious. I wonder if anyone has a nice and clear ...
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2answers
113 views

Degree of closure of $\mathbb{Q}_p$

In order to prove that algebraic closure of $\mathbb{Q}_p$ is infinite, I took the polynomial $x^n-p$ with $n>1$ over $\mathbb{Q}_p$ to show that this eqaution has no solution for infinite cases to ...
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1answer
199 views

Firoozbakht's conjecture solution?

Not so much an question as adding another level to the same question as Ratio of logarithmic primes. (See answers, same as here.) The Firoozbakht's conjecture (1982) is equal to: $$(p_{n+1})^{n} ...
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3answers
71 views

How prove this irrational $x\in(0,1)$,then have $0<x-\sum_{i=1}^{n}\frac{1}{p_{i}}<\frac{1}{n!(n!+1)}$

show that: for any irrational $x\in(0,1)$,and positive integer $n$,there exsit positive integer $p_{1},p_{2},\cdots,p_{n}$ where $$p_{1}<p_{2}<\cdots<p_{n}$$ such ...
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2answers
180 views

Is the ring of integers of a local field an open subgroup?

I apologize if my question is a bit naive, but I don't have much experience in number theory and sometimes get very confused. Suppose $K$ is a non-archimedean local field (essentially a completion of ...
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4answers
470 views

How find this $x,y$ such this equation $x^x+x=y!$

Find all pairs of positive integers $(x,y)$,such $$x^x+x=y!$$ I find the $$(x,y)=(1,2)$$ is such it. and $$(x,y)=(2,3)$$ I think this equation have other roots. and maybe use inequality to ...
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4answers
132 views

Prove that $(2m+1)^2 - 4(2n+1)$ can never be a perfect square where m, n are integers

I could prove it hit and trial method. But I was thinking to come up with a general and a more 'mathematically' correct method, but I did not reach anywhere. Thanks a lot for any help.
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1answer
76 views

infinite primes $p\equiv1\pmod n$ without cyclotomic polynomial

Without cyclotomic polynomial, is there an elementary proof of the following: for each integer $n>1$, there are infinitely many primes $p$ such that $p\equiv1\pmod n$ ? please don't refer to ...
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5answers
67 views

For any prime $p>3$ show that 3 divides $2p^2+1$

Does anyone know how to show this preferable without using modular For any prime $p>3$ show that 3 divides $2p^2+1$
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1answer
96 views

How prove this sequence $S_{n}=[2^n\cdot \sqrt{2}],n\in N$ contains infinitely many composite numbers

define sequence $$S_{n}=[2^n\cdot \sqrt{2}],n\in N$$ show that $\{S_{n}\}$contains infinitely many composite numbers where $[x]$ is the largest integer not greater than $x$ my try: since ...
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4answers
130 views

Show that if $p$ is a prime number $> 3$ then $24 \mid p^2-1$ [duplicate]

Hi guys can someone help me with this ?(Without using Modular arithmetic) Show that if $p$ is a prime number $>3$ then $24$ $\mid$ $p^2-1$
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0answers
40 views

Is there a general way to have a polynomial in two variables over C (a plane curve) be irreducible?

Is there a general way to have a plane curve be irreducible? If the curve $C \in \mathbb{C} [x,y]$, would it be sufficient for it to factor into linear terms? What about if I have an equation of the ...
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1answer
57 views

Explain theorem in Number theory

can some one explain with a clear example this theorem for me, Let ($A_1$, $A_2$, $A_3$,..., $A_n$) be integars and $p$ a prime number. if $p|(A_1A_2A_3...A_n)$ then there exist some $1 \leq k \leq ...
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3answers
86 views

If $2$ divides a number $a$, does $2^n$ divide $a$ ? $n$ is any integar

If $2$ divides a number $a$, does $2^n$ divide $a$ ? $n$ is any integer. This seems to be true for me, but I just want to make sure it applies for all numbers. example if a = 137 2 does not divide ...
4
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1answer
63 views

Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$?

Let $\mathbb{Q}_p$ be the set of all p-adic numbers and $\mathbb{Z}_p$ the set of all p-adic integers. Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$? Thank you very ...
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1answer
86 views

How to show that $\mathbb{Q}_p^*$ is totally disconnected?

Let $\mathbb{Q}_p$ be the field of p-adic numbers and $\mathbb{Q}_p^*$ the set of invertible elements in $\mathbb{Q}_p$. How to show that $\mathbb{Q}_p^*$ is totally disconnected? Thank you very ...
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1answer
69 views

Concerning types of square-free numbers and comparing sizes of their subsets.

Call a square-free a 2-prime if it has exactly two prime divisors. Call a square-free a 3-prime if it has exactly three prime divisors,etc. Does there exist an integer sufficiently large N such that ...
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1answer
53 views

Showing that $\lambda(n)|\phi(n)$ where $n$ is a positive integer.

Let $n$ be a positive integer. Let $\phi(n)$ be the Euler phi function and $\lambda(n)$ be the Carmichael lambda. I know that $\phi(n)=\phi(2^{t})\phi(p_1^{t_1})\cdots\phi(p_n^{t_n})$. I also know ...
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1answer
126 views

The number of rational solutions to the cubic analogue of Pell's equation

The equation: $x^3+dy^3+d^2z^3-3dxyz = 1\tag{1}$ is the cubic analogue of Pell's equation. For a given positive integer $d$ not a perfect cube there is essentially one solution in positive integers ...
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0answers
168 views

Primitive Roots modulo a prime

I'll give the question, and then I'll describe the headway I've made so far. "Let $p$ be an odd prime, and suppose that $q=2p+1$ is also prime. Prove that if $a$ is incongruent to $1$ and ...
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2answers
55 views

Concerning types of square-free numbers.

Call a square-free number a 3-prime if it is the product of three primes. Similarly for 2-primes, 4-primes , 5-primes, etc. Are there two consecutive 3-primes with no 2-prime between them?Are there ...
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3answers
22 views

Solving a system of linear congruences with a small mistake

I having trouble solving the system of linear congruences: $x \equiv 1 (mod 2)$ (a) $x \equiv 1 (mod 3)$ (b) so what i do is from (a) $x = 2y + 1$ and into (b) $2y + 1 \equiv 1 (mod 3)$ so $2y ...
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1answer
58 views

Concerning what is between two consecutive squares.

Is there a squarefree ( with two prime divisors) betweem any two consecutive squares?
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2answers
66 views

Describe all integers a for which the following system of congruences (with one unknown x) has integer solutions:

$$x\equiv a \pmod {100}$$ $$x\equiv a^2 \pmod {35}$$ $$x\equiv 3a-2 \pmod {49}$$ I'm trying to solve this system of congruences, but I'm only familiar with a method for solving when the mods are ...
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1answer
86 views

Find all solutions to the Diophantine equation $2x+3y =1$.

How to find all the solutions to the Diophantine equation $2x+3y =1$. My professor didn't explain to us how to do this.
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2answers
919 views

$r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root

question is as follows: Let $p$ be a prime with $p \equiv 1 \mod 4$, and $r$ be a primitive root of $p$. Prove that $-r$ is also a primitive root of $p$. I have shown that $-r^{\phi(p)} \equiv 1 ...
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1answer
66 views

Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
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1answer
77 views

Problem on Number of Quadratic Residues

We have two primes $p,q$ and an integer $a$ such that $$\gcd(a,pq)=1$$ How to prove that for the following congruence $$x^2 \equiv a \mod pq$$ either there will be $4$ solutions or $zero$ solutions. ...
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2answers
466 views

elementary proof that infinite primes quadratic residue modulo $p$

$p \gt 2$ is a prime, then there are infinite primes $q$ such that $q$ is a quadratic residue modulo $p$. With Dirichlet's theorem on arithmetic progressions, the problem is easy! How about ...
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4answers
123 views

Number theory with positive integer $n$ question

If $n$ is a positive integer, what is the smallest value of $n$ such that $$(n+20)+(n+21)+(n+22)+ ... + (n+100)$$ is a perfect square? I don't even now how to start answering this question.
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3answers
66 views

Find all natural numbers $x$ such that …

Find all natural numbers $x$ such that product of their digits is equal to $x^2-10x-22$ I've already found one such number it is $12$ because $(12)^2-10\cdot12-22=2$ and $1\cdot2=2$ but I don't know ...
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1answer
165 views

Modular Form, but Cusp Form in Disguise

Suppose I have a holomorphic modular form $f \in M_k(\Gamma_0(N), \chi)$ with $k \in \mathbb Z^+$ and $f = \sum_{n=1}^\infty a(n)q^n$. Considering the $q$-expansion, one may suspicious that $f$ is ...
5
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2answers
162 views

Primality of the number of obtained by concatenating the n consecutive digits

Let $f_n$ be the number obtained by concatenating the first $n$ numbers (in base 10). For example $f_1 = 1, f_3 = 123$ and $f_{13} = 12345678910111213.$ Now if $n$ is even or divisible by $5$ then ...
3
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1answer
72 views

Is it true that $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$?

Let $p$ be a prime number, are the following statements true? 1.Quadratics of the form $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$. And all such primes ...