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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Different Ideals and their relationship to tame/wild ramification

I am looking at Marcus' book "Number fields", precisely at exercises that lead to Hilbert's formula $k= \sum_{m\ge0} (|V_m|-1)$; that can be applied in turn to prove that, if $L/K$ is a Galois ...
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113 views

Strategies for solving rational Diophantine equations

Are there any strategies for solving Diophantine equations where the solutions can be any rational number, not just an integer, besides substituting $x=p/q$ and $y=r/s$, with $p,q,r,s$ integers with $\...
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25 views

Count pairwise integer modulo

Assume $N$ and $M=(N-1)/2$ are both prime. For any $L(L\leq N)$ different integers $1\leq i_1<i_2<\ldots<i_L \leq N$, denote $A_m(i_1,\ldots,i_L)$, $1\leq m \leq M$ to be the number of ...
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40 views

Number of distinct prime factors of $a^n+b^n+c^n$

This question is a generalization of this other one. Problem: Given a constant $k$ and distinct positive integers $a,b,c$, prove that there exist an integer $n>1$ for which the number of distinct ...
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37 views

Generalized elliptic curves over cusps and orbits of $\mathbb{Q}\cup\infty$

In the post http://mathoverflow.net/questions/51147/what-objects-do-the-cusps-of-modular-curve-classify, it says that the fibers over the cusps of a modular curve are n-gons. Wikipedia (https://en.m....
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30 views

Chinese Remainder theorem clarification needed

I'm trying to solve $p\equiv a\pmod{7}$ and $p\equiv b\pmod{4}$ $m_1=(4)^{-1} \pmod 7$ $m_2=(7)^{-1} \pmod 4$ I need to find $m_1'$ and $m_2'$ which I assumed to be the inverse of $m_1$ and $m_2$ ...
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Operation on primes with power $\geq 3$

Question. Can it be shown that if $m, n$ are odd integers $\geq 3$, and with no common factor, then $m^z - n^y = (2^x).K$ where $y, z$ are integers $\geq 3$, $K$ is an odd integer $\geq 3$, and $x$ is ...
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Additive combinatorics in Hamming weights of addition of numbers modulo $2^n$ with prescribed Hamming weight

I wonder if anyone could point me to a reference about the following type of combinatorics problem: Fix $n $. For an integer $k \in [n] = \{1, \ldots, n\}$, let $A(k)$ be the set of integers in $[0, ...
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Bounding the size of integer solutions of curves $x^{r}+y^{s}=z^{t}$

I was reading Poonen's paper "Twists of X(7) and primitive solutions of $x^{2}+y^{3}=z^{7}$ (available here) where he describes how he found all primitive solutions of the above curve. A solution is ...
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22 views

How much countable additivity does asymptotic density have?

For $A\subseteq \mathbb N = \{1,2,3,\ldots\}$ let $d(A)$ be the "density" defined by $$ d(A) = \lim_{n\to\infty} \frac{|A \cap \{1,\ldots,n\}|} n \tag 1 $$ whenever that limit exists. This is ...
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Prove this inequality for sufficiently large $n$

Prove that the above inequality holds for sufficiently large $n$: $$\pi(2n) - \frac{3}{2} \pi(n) \ge O\left(\frac{\ln n}{(\ln \ln n )^2}\right)$$ $\ln n$ denotes to natural logarithm and $\pi(n)$ is ...
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number theory RSA encryption problem 2

Let $p$ and $q$ be distinct prime numbers and $N =pq$. Do multiplication $\mod N$. Define $G $= {natural numbers less than $pq$ that are relatively prime to $pq$} and $s =(p-1)(q-1)$ . Then $(1)$ $m^...
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Help with contradiction in proof of Euler's Theorem

The theorem states: if $n\in\mathbb{N}$ and $(a,n)=1$ then $$a^{\varphi (n)}\equiv 1\bmod n$$ Where $\varphi$ is the Euler-Phi function Take $a\in\mathbb{Z}$ such that $(a,n)=1$. Consider $\mathcal{U}...
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25 views

Convergents of continued fractions

Let $d$ and $m$ be positive integers such that $d$ is not a square and such that $m\leq\sqrt{d}$. I want to prove that if $x$ and $y$ are positive integers stafisfying $x^2-dy^2=m$ then $x/y$ is a ...
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33 views

A Fact Stated in Davenport's Multiplicative Number Theory

In his text Multiplicative Number Theory on page 9, Davenport mentions that another means of expanding the L-function is known and then mentions the fact that, $$ \mathcal{F} \sum_{n=1,n \; odd} \...
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46 views

Number Theory: Find a primitive root of $13^{901}$ and find a complete set of primitive roots of $13$

I solved this problem: Find a complete set of mutually incongruent primitive roots of $13$. I know that there are $\phi(\phi(13))=4$ primitive roots of 13, which are $2,6,7,$ and $11$. However, I ...
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36 views

How many divisors does $n\alpha$ have?

if $$\frac {p^\alpha - 1}{p - 1}= 2^n \hspace{5 mm} n,\alpha \in \mathbb{N} \hspace{5 mm} p\in\mathbb{P}$$ How many divisors does $n\alpha$ have? I did make some attempts but most of them were ...
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Solve $a^n - b^n = 8$ with $a,b \in \mathbb{Z}$ and $n \in \mathbb{N}_{\geq 2}$.

I already solved the question myself, but there is something bottering me. In the exersice is told "Solve, by easy estimations". I couldnt find a boundary for $n$ or something. I started with $a^n = 8 ...
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80 views

A problem of periodic functions , the greatest common divisor and a lattice

I am trying to solve the following problem. If $\psi(s) = \frac{s(s-1)}2$. I write $f(s,k) = (\psi(s),\psi(s-k))$, where $k$ is a fixed positive integer. Let $K$ be the image of $f_{s,k}$. If $s>3$...
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Impossible form of a triangular number

Show that there are no positive integers $t,i,j$ with $j>i$ such that: $\displaystyle \frac{t(t+1)}2=\frac{2i(j-i)j(j+i)}3$ If possible provide an elementary proof. I believe the statement is ...
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What determines the number of families to $1-4x-4(1-x^2)z = w^2$?

This is related to this post. First, we have, Theorem: "If $w_0, z_0$ is a solution to, $$1-4x-4(1-x^2)z = w^2\tag1$$ then, $$w = w_0+2(x^2-1)n$$ $$z = z_0+w_0\,n+(x^2-1)n^2$$ is also a ...
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60 views

Collatz Conjecture: Literature on Convergence

Does anyone know of a paper showing that if all n converge, they must converge to unity for n>0. Else, any literature related to convergence properties would be appreciated. Thanks, Jordan
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Divisibility by primes in $\mathbb{Z}/n\mathbb{Z}^{\times}$

Fix $N$ such that $N\approx 2^{80}$. Let $a,b$ be randomly chosen positive integers such that $a,b<N$ and both $a$ and $b$ are coprime to $N$. I want to show that the odds that $gcd(a,b)=1$ are ...
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134 views

Is there a proven way to calculate the entry point(first occurence) of a factor m, in the Fibonacci sequence?

I saw a comment at the OEIS website for the sequence of entry points, of Fibonacci factors. https://oeis.org/A001177 It referenced a paper by Mark Renault in 1996, with the quote from OEIS: http://...
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34 views

Fourier series concerning Gibbs constant and the divisor function.

It is quite a remarkable function I found. It seems, though, that I may be staring at something trivial, which is hopefully not the case. I would like some opinions. The function is $$f(b)=\lim\...
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Hensel's Lemma Argument

Let $f(X)$ be a polynomial with integer coefficients. Let $p$ be a prime number. Recall that ord$_p$$ : Z → N∪ {∞}$ is the function such that ord$_p(0) = ∞$, $n ≡ 0 ($mod $p^{ord_p(n)})$, and $n\...
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Give a sufficient condition for the general monic quadratic polynomial $f(X)=X^2+bX+c∈\mathbb Z[X]$ to have solutions in $\mathbb{Z}/p^n\mathbb{Z}$

Let $p$ be a prime number. Combine Hensel’s lemma and quadratic reciprocity to give a sufficient condition for the general monic quadratic polynomial $$f(X) = X^2 +bX +c ∈\mathbb Z[X]$$ with integer ...
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Importance and Applications of Cuban Primes

Are there any applications of cuban primes, or are they only considered to be within the realm of pure mathematics? Is there anything significant about them specifically? Would more research into ...
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Is every primorial number squarefree?

Is every primorial number ( a number of the form $p$#$\pm 1$ ) squarefree ? According to my calculation, there is no prime $q\le 270,000$, such that $q^2$ can be a divisor of $p$#$-1$ or $p$#$+1$. ...
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Prove that the canonical p-adic expansion of $a\in\mathbb{Q}_p$ terminates

I am having trouble getting started with the following problem: Prove that the canonical p-adic expansion of $a\in\mathbb{Q}_p$ terminates (so $a_i = 0$ for all $i \geq N$) if and only if a is a ...
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Looking for an elegant proof of why there is only ever a single intersection point of the nullclines for a 2D ODE system

I have the following ODE system $$ \dfrac{d \ell}{dt}=\sigma_{\ell} - \mu_{\ell} \dfrac{M \ell}{1+\ell}-d_{\ell}\ell \\ \dfrac{dM}{dt}=\dfrac{\ell}{\beta+\ell}+\sigma_M \dfrac{M \ell}{1+\ell}-M $$ ...
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20 views

Lower bound on number of relatively prime pairs

Consider the sets of consecutive positive integers:\ $A = \{ a,a+1,...,a+n-1 \}$,\ $B = \{b, b+1, ..., b + m - 1\}$.\ where $n$, $m \in \mathbb{Z}$ with $3 \leq n \leq m$. Is there a formula for a ...
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68 views

Am I pretty close to proving that e is irrational?

Show that $e=1+1/1!+1/2!+1/3!+…$ is an irrational number. Hint: show that, for all positive integers $p$, $0<p![e−(1+1/1!+…+1/p!)]<1$. Then conclude that $e$ cannot be a ratio of two integers q/...
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On chinese remainder theorem

Given $a\equiv x \bmod r_1$ and $b\equiv x\bmod r_2$, we can construct $x$ from $$x=a r_2 [r_2^{-1}]_{r_1} + b r_1 [r_1^{-1}]_{r_2}$$ where $(r_1,r_2)=1$ and $[r_2^{-1}]_{r_1}$ is residue class of $...
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19 views

Suppose we have a polynomial over the integers with a non-zero leading coefficient over mod p.

Suppose we have a polynomial over the integers with a non-zero leading coefficient over $\mod p$. Suppose $r$ is a zero of $f(r)$ is congruent to $0 \mod p$, show there exists polynomial $g(x)$ such ...
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37 views

If $\gcd(Z,\sigma(Z))=1$ and $1<N=Z\sigma(Z)$, is $N$ always friendly?

This question is a generalization / offshoot of this earlier MSE post: If $\gcd(Z,\sigma(Z))=1$ and $1<N=Z\sigma(Z)$, is $N$ always friendly? Here, $\gcd(a,b)$ is the greatest common divisor ...
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Why is proving that $10$ is solitary considered very difficult?

The title says it all. We denote the sum of the divisors of $x$ by $\sigma(x)$. The ratio $I(x)=\sigma(x)/x$ is called the abundancy index of $x$. If $I(m)=I(n)$, then $\{m,n\}$ is called a ...
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48 views

Is $p(p + 1)$ always a friendly number for $p$ a prime number?

Let $\sigma(x)$ denote the sum of the divisors of $x$. We call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A positive integer $N$ is friendly if there exists a positive integer $M \...
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Can a non-trivial factor of a strong Fermat-pseudoprime always be found efficiently?

Suppose, $N$ is a composite Fermat-pseudoprime to base $a$ : $$a^{N-1}\equiv 1\ (\ mod \ N)$$ If $N$ is NOT strong Fermat-pseudoprime to base $a$, a non-trivial factor of $N$ can be found ...
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Does Linnik's approximation to Goldbach's problem also work for the power of 3, 5, 7, etc ?

Linnik proved in 1951 the existence of a constant K such that every sufficiently large even number is the sum of two primes and at most K powers of 2. Roger Heath-Brown and Jan-Christoph Schlage-...
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37 views

Prove that a set of numbers can never consist of prime numbers

I noticed that for any integer $n>0$, $14^n+11$ would not be prime for up to $n=4$. Does this hold for all $n$? Is there a way to prove this or is this something that only works for some $n$?
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No closed form for $\sum_{n\in P} \frac{1}{n^2}$

I think that I can say with a fair amount of assurance that $$\sum_{n\in \mathcal P} \frac{1}{n^2}$$ has no closed form (assuming that $\mathcal P$ represents the full set of primes) I currently know ...
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40 views

Locating a pdf of certain paper

One of the references in the book Opera de Cribro by Friedlander and Iwaniec is: J. Friedlander and H. Iwaniec, A polynomial divisor problem, J. Reine Angew. Math. 601 (2006), 109-137 Does anybody ...
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72 views

Index of a sublattice

Assume that $\Lambda \subset \mathbb{Z}^2$ is an integer lattice of rank $2$ and define for each integer $k$, $$\Lambda(k):=\Big\{(x_1,x_2) \in \Lambda: k\mid (x_1,x_2)\Big\}.$$ What is the index of $[...
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57 views

Thoughts on Mersenne numbers

Mersenne numbers are numbers of the form $2^n-1$. Let us use the standard symbol $M_n$ for them. I am interested here in opinions about methods for obtaining the answers of these two questions: ...
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39 views

Is there a Carmichael-number divisible by $3\times 5\times 17=255$?

I am searching a Carmichael-number of the form $N=3\times 5\times 17\times k$. $N$ must have the form $N=4080m+3825$ because $N$ must satisfy $N\equiv 0\ (\ mod\ 255\ )$ and $N\equiv 1\ (\ mod\ 16)$. ...
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28 views

Textbooks on transcendence theory

Is there a nice, modern textbook (some lecture notes or survey would do, too) that covers the main results and methods from transcendence theory? Ideally, it should also have some good exercises. So ...
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20 views

Is it possible to write identity: $(x(y^2-z^2)-y).(u(v^2-w^2)-v))=a(b^2-c^2)-b$?

Is it possible to write identity for $$ \{x(y^2-z^2)-y\}.\{u(v^2-w^2)-v)\}=a(b^2-c^2)-b $$ in integers similar to the identity $$ (x^2+y^2)(u^2+v^2)=(xu+yv)^2+(xv-yu)^2 $$ If possible, what can we ...
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38 views

what is number of unique positive divisor of an integer n?

I am working on my number theory assignment and we have this theory that states that sum of Euler's totient function of all positive divisor equals the number itself this is just a little back ...
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50 views

Prove that there is no integer $k$ with $P(k)=8$

Let $P(x)= x^n + a_{n-1}x^{n-1}+...+a_1x+a_0$be a polynomial with integral coefficients. Suppose that there exists four distinct integers $a$, $b$, $c$, $d$ with $P(a)=P(b)=P(c)=P(d)=5$. Prove ...