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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Prime Numbers and a Two-Player Game

Alex and Beth is playing the following game. Initially, there are $n\in\mathbb{N}\cup\{0\}$ marbles on a table. Alex and Beth alternatively pick out some marbles out of the table, with Alex going ...
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0answers
5 views

Properties of a Semi-modulo! operation

Let $A$ be an integer with its representation in base $p^n$ ($p$ may be prime number but not necessarily) described as: $$A=(a_ma_{m-1}\ldots a_1a_0)_{p^n}$$ We know $A\equiv (a_n\ldots ...
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0answers
13 views

How is named this property of zero parity?

How is named then this property of zero parity? Numbers parity Any number, except zero, multiplied twice, is an even number. A pair has two elements. Demonstration: One multiplied ...
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1answer
29 views

Inequality involving a product over the primes

Is there someone who is able to prove the following statement? $$\prod_{m=1}^n \dfrac{p_m-1}{p_m} \leq \dfrac{1}{\ln(n)}$$ for all integers $n >1$ where $p_m$ is the $m$-th prime number.
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2answers
109 views

Is $a=\frac{1992!-1}{3449\times 8627}$ a prime number?

Is $a=\dfrac{1992!-1}{3449\times 8627}$ a prime number ? This is a natural follow-up to that recent MSE question We know that $a$ has $5702$ digits and no prime divisor $<10^6$.
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0answers
39 views

What is the inverse function of gcd?

Let $a,x,c \in\mathbb{Z}$. If $\gcd(a,x)=c$ where $a, c$ are constants and $x$ is a variable, then what values can $x$ take and how to find those values ?
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1answer
28 views

Is there an isomorphism $\mathbb{C}_p \cong \mathbb{C}_q$ for primes $p \neq q$?

Let $p \neq q$ be distinct primes. Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}_q$? Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}$? If such an isomorphism exists, given ...
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34 views

A question on the inequality $\bigl(\pi(x+y)\bigr)^2<4\pi(x)\pi(y)$

From the answer of this post it seems highly probable that the following problem can be proved, Show that for all sufficiently large $\min(x,y)$ we will have, ...
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0answers
11 views

Maximal $n$ such the the additive partition with a given product is unique.

Given $n$, there are many tuples with $a + b + c = n,a < b < c$. For large $n$, different tuples may give the same products. E.g. $2+8+9=19=3+4+12,2\times8\times9=144=3\times4\times12$. What is ...
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1answer
32 views

Congruence using extended GCD

$$\eqalign{ x &\equiv 5 \mod 15\cr x &\equiv 8 \mod 21\cr}$$ The extended Euclidean algorithm gives $x≡50 \bmod 105$. I understand now that if we combine the two it implies ...
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2answers
45 views

Why is this congruence true?

$$\eqalign{ x &\equiv 5 \mod 15\cr x &\equiv 8 \mod 21\cr}$$ The extended Euclidean algorithm gives $x≡50 \bmod 105$. How/why? I am trying to understand how this is true when ...
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1answer
32 views

$2$-adic sequence converging to $\sqrt{-7}$.

I am trying to construct a sequence in $\mathbb Q_2$ that is formed of rational numbers and converges to $\sqrt{-7}$, to prove that $(\mathbb Q, |\cdot|_2)$ is not complete. My lecturer stated that ...
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0answers
29 views

A residue question in integers

Given $N\in\Bbb N$, is it possible to find $9$ positive integers $A_j,N_i$ with $j\in\{1,2,3\}$, $i\in\{1,2,3,4,5,6\}$ such that following holds? $(1)$ $N\log N < A_j < cN\log N$ at every $j$ ...
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1answer
27 views

Prove that $\sum_{t \vert n} d^3(t) = (\sum_{t \vert n}d(t))^2$ for all $n \in \mathbb{N}$ [duplicate]

here $d(n)$ counts the number of positive divisors of $n$. I've tried 2 things: Using Bell series. But then again it just showed me that the bell series of the square of a function is not the ...
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0answers
19 views

Is there a solution to this problem on Fermat's Quotient?

We define Fermat's Quotient as $q_a = (a^p-1)/p \pmod p $ where $p$ is a prime greater than $2$. How will you prove that the only solutions of the equation $q_a=0$, $q_b=0$ and $q_{a+b}=0$ where ...
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0answers
18 views

Need my CRT work spot-checked

So I have a bunch of equations that look like this: $$k + tx \equiv a \bmod m$$ Where $t$ is the common variable I am solving for among the equations (each equation may have different values for ...
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1answer
28 views

Why does this imply a congruence does not exist?

Consider $$kx \equiv a \bmod M$$ Where $x,a,M$ are known, solving for $k$. Let $g = \gcd(M,x)$. Why is it the case that if $g$ does not divide $a$, there is no solution?
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1answer
92 views

Possible proof of Fermat's Last Theorem for prime exponents greater than 2

I would appreciate if someone could check my attempt in proving the Fermat's Last Theorem for prime exponents greater than $2$. Firstly, let's prove a couple of lemmas which state that sum or ...
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0answers
34 views

Why is it so hard to find a generating function for Somos' sequence?

The sequence is $\{1,2,12,576,1658880,\dots\}$. The $n$th number is obtained by squaring the $(n-1)$-th number and multiplying by $n$. So we start with $a_1=1$, $a_2=1^22=2$, $a_3=(1^22)^23=12$. In ...
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0answers
50 views

rationality of $\ell$-adic representation attached to an elliptic curves

Let $E$ be an elliptic curves defined over a number field $K$. Consider the $\ell$-adic representation attached to $E$ $$ \rho_{\ell}:\mathrm{Gal}(\overline{K}/K) \longrightarrow ...
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1answer
60 views

Question about Mersenne numbers

We don't have any proof for Mersenne conjectures, but is it true that there exist infinitely many primes $p$ such that $2^p-1$ is not prime?
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13 views

$S=\{1\le a \le n:(a,n)=1,a^{n-1}\not\equiv 1\pmod n\}$, $T=\{1\le b \le n:(b,n)=1,b^{n-1}\equiv 1\pmod n\}$ iwth composite and prime numbers

I have two sets with $n>2$ natural number: $S=\{1\le a \le n:(a,n)=1,a^{n-1}\not\equiv 1\pmod n\}$ $T=\{1\le b \le n:(b,n)=1,b^{n-1}\equiv 1\pmod n\}$ Can anyone explain me if there are prime ...
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1answer
83 views

The existential theory is undecidable

Lemma 1. For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and only if $x$ divides $1$ and ...
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0answers
47 views

Are logarithms of prime numbers algebraically independent?

From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero. Has it been proved that the set of all natural ...
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2answers
50 views

All natural numbers $m, n$ such that $m = \sqrt{\frac{1}3A^2 - 3n^2}$ [on hold]

I have $m = \sqrt{\frac{1}3A^2 - 3n^2}$. A is a known integer. How do I find all solutions for what m and n are if both m and n are naturals (round positive numbers)
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1answer
404 views

Is $1992! - 1$ prime?

Consider the factorials, defined inductively by $1! = 0! = 1$ and $n! = n\cdot(n-1)!$ for $n \geq 2$. Question: Is $1992!-1$ a prime number? The question is from a book, maybe is contest math ...
3
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1answer
43 views

Proof of Cohn's Irreducibility Criterion

I was looking for an elementary (or involving introductory level abstract algebra/analysis) proof of Cohn's Irreduciblity Criterion: If $$ a_0, a_1, \dots, a_n \in \Bbb{Z} $$ and $$ 0 \le ...
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3answers
34 views

Finding the largest square divisor of a number

My calculator has the option of representing square roots in the form of $a\sqrt b$, when $a$ is maximal. It works for very large inputs within seconds, and I wonder how it's being done.
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1answer
35 views

NumberTheory: Proof or disproof the following. Dividing and adding

Proof or disproof the following. Let $N \in \mathbb{N}$ be a natural number. If we divide the digits of $N$ with preserving the order and adding them together we will get a digit ( a number less ...
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1answer
102 views

Is the real number $\sqrt{6}$ in $\mathbb{R}$ equal to the 5-adic number $\sqrt{6}$ in $\mathbb{Q}_5$?

My question is as in the title. That is, consider solving the equation $x^2-6=0$ in $\mathbb{R}$ and in the 5-adic field $\mathbb{Q}_5$ respectively. We obtain one $\sqrt{6}\in\mathbb{R}$ and one ...
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1answer
26 views

Asymptotic probability that two integers are coprime

I'm having difficulty with a number-theory-type exercise. Could you provide assistance with computing the asymptotic probabilities that two integers are coprime (both integers tending to $\infty$), ...
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0answers
43 views

Is the Green-Tao theorem valid for arithmetic progressions of numbers whose Möbius value $\mu(n)=-1$?

I am reading the basic concepts of the Green-Tao theorem (and also reading the previous questions at MSE about the corollaries of the theorem). According to the Wikipedia, the theorem can be stated ...
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2answers
37 views

Congruence rules when solving equation

I am trying to solve the following congruence problem. 980x ≡ 1500 mod 1600 The steps I came up with were as follows: 980x ≡ 1500 mod 1600 49x ≡ 75 mod 80 (Divide by 20, gcd(20, 1600) = 20 so 80 = ...
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46 views

Prove that $n \in \mathbb{Z}^\star \Leftrightarrow n \mid 1$ and $n-1 \mid 1$ or $n+1 \mid 1$, and $(x-1)/(t-1) \equiv n \pmod {t-1}$

I want to prove the following lemma: Let $F[t,t^{-1}]$ be the ring of the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$ and assume that the characteristic of $F$ is zero. ...
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0answers
24 views

On Period of Linear Recurring Sequences modulo $P^e$

If a sequence $ X_0,X_1,X_2,\ldots$ is defined in terms of an initial set $ X_0,X_1,X_2,\ldots ,X_{k-1} $ by the recurrence relation $$ X_{n+k}= ...
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1answer
14 views

Comparison of arbitary conway chains (in particular a chain with $m$ $m's$) to $f_{\omega^2}(n)$

Wikipedia describes the Conway chained arrow notation and the fast growing hierarchy. I learnt that the function $f(n):=\large f_{\omega^2}(n)$ has the same growth rate as the function ...
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1answer
33 views

sum of the series of certain form close to Fermat's numbers

My question is: What is the sum of reciprocals of the numbers $2^{2^n}$. If we achieve this we will be able to give a good bound for the sum of reciprocals of Fermat's numbers i.e. $(2^{2^n})$+1.
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1answer
20 views

Why the action of $\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ on $\overline{\mathbb Q}_p$ restricts to $\overline{\mathbb Q}$?

Let $\overline{\mathbb Q}$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$ and chose an algebraic closure $\overline{\mathbb Q}_p$ for $\mathbb Q_p$. The embedding $\mathbb Q \hookrightarrow ...
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35 views

Motivation behind the Archimedean norm on number fields .

It is easy to justify the use of non-archimedean norms on number fields from an "inside view" as arising from the prime ideals and are therefore clearly useful a priori. However, it seems to me that ...
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6 views

Does $F\otimes G\in\mathcal{M}$?

Let $\mathcal{M}$ be the class of automorphic L-functions which belong to the Selberg class. Let $F$ and $G$ be elements of this class, and define $F\otimes G$ by $a_{p}(F\otimes G)=a_{p}(F).a_{p}(G)$ ...
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2answers
16 views

Does changing the value of x change the number of solutions?

So I have the equation: $$-C<2n+x<C$$ Where $$n ∈ Z$$ $$C ∈ R$$ $$-1<x<1$$ My question is, for a given value of C, do the same number of values for n always exist, regardless of the ...
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0answers
43 views

Which permutations of $\mathbb{C}$ commute with the Riemann zeta function?

I'm trying to figure out whether the permutations of $\mathbb{C}$ which commute with the Riemann $\zeta$ function are necessarily continuous or not. Obviously both the identity and the complex ...
2
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1answer
36 views

Generalization of Erdos-Selfridge

Consider the equation $P(x)=y^d$ where $d \geq 2$ is an integer and $P$ can be written $P(x)=c(x-r_1)(x-r_2)\ldots (x-r_t)$ where $c$ and all the $r_i$ are integers not all equal (some of them can be ...
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0answers
69 views

On a unique(?) binomial property of $3003$

Given the triangular number, $$T_k = \frac{k(k+1)}{2}$$ and remembering that, $$\binom{n}{m}=\binom{n}{n-m}$$ Excluding $a_0=1$, we then have the six-fold (at least) equalities, $$\begin{aligned} ...
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21 views

$x-y^4= LCM(x, y)$ [duplicate]

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with the least common multiple, but other than that, the textbook gave no hints ...
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1answer
56 views

Pairs of integers with gcd equal to a given number

Given integers $N$ and $D$, find how many pairs of integers $(i, j)$ such that $1 \le i \le j \le N$ have the greatest common divisor exactly $D$. I know it involves Mobius inversion somehow, but I ...
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1answer
45 views

Describe a fast (polynomial time)algorithm who takes as input the elements $g^a,g^b$ and gives as output the element $g^{a \cdot b}$

Let $q$ prime number, $G$ a cyclic group with order $q$ and $g \in G$. Suppose that you have an algorithm $A$ who takes input the element $g^a$ of $G$ and gives as output the element $g^{a^2}$. ...
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86 views
+50

A lower bound for an arithmetic function

Let $N \in \mathbb{N}$ such that $\phi(N) \sim N$, where $\phi$ is the Euler's totient function. Let $A \subset [N] := \{1, 2, \ldots, N\}$. For $n \in \mathbb{N}$ define the function $$ C_A(n) = \#\{ ...
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55 views

System of equations to solve this nested radical.

The nested radical $$1.75793\approx\sqrt{1+\sqrt{2+\sqrt{3+\cdots}}}$$ has yet to be given a closed form. However, nested radicals of the form, $$\sqrt{A+B\sqrt{A+B\sqrt{A+\cdots}}}$$ have the ...
3
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2answers
61 views

The algebraic closure of $\mathbb{Q}_p$

I am trying to explain why $\mathbb{Q}_p^{\text{alg cl}}$ is an infinite field extension of $\mathbb{Q}_p$ (unlike $\mathbb{C}/\mathbb{R}$ which has deg 2). Does the following argument work out... ...