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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14 views

Chinese Remainder Theorem example

$$x = 4 \bmod 18$$ $$x = 52 \bmod 96$$ $$x = 6 \bmod 20$$ I've been trying to put together a program to solve the CRT (even when things may not be coprime, so I am just trying to go over cases that ...
0
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0answers
8 views

Proofs needed for observations regarding prime-partitionable numbers.

Given $P_1 =${$p_{1a}, p_{1b}$}, $p_{1a}$ and $p_{1b}$ are odd primes and $p_{1a}< p_{1b}$, it appears that, if $p^2 = k_{1a}*p_{1a} + 1 = p_{1b} + p_{1a}$ for $k_{1a}$ odd and $1 < k_{1a} ≤ ...
2
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0answers
16 views

solutions for Diophantine equation for ${k_1} + 2{k_2} + 3{k_3} + … + n{k_n} = n$

Consider the Diophantine equation of the form ${k_1} + 2{k_2} + 3{k_3} + .... + n{k_n} = n$. For a given $n$, how can I obtain a general solutions for the given equation?
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2answers
60 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth is playing the following game. ...
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0answers
12 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
23 views

How is named this property of zero parity? [on hold]

How is named then this property of zero parity? NOTE: I'm worrying because: Wikipedia started a project Wikipedia Zero. They named it because zero is considered "an even number"? ...
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2answers
80 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.
5
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2answers
112 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
41 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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0answers
37 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
12 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 ...
1
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1answer
36 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
46 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 ...
5
votes
1answer
36 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
20 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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votes
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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votes
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 ...
2
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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 ...
4
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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 ...
4
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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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0answers
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 ...
3
votes
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)
21
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1answer
412 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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votes
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 ...
3
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1answer
104 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 ...
1
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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 $\infty$), ...
1
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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 ...
3
votes
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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0answers
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}= ...
1
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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 ...
4
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0answers
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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0answers
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 ...
3
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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
votes
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 ...
4
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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} ...
2
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0answers
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 ...
0
votes
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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votes
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 $g^{a^2}$. ...