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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Sum of reciprocal of primes in arithmetic progression

In http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf on page 6 (top) the author states that: $$ \sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} = \frac{\log \log x}{\phi(l)} + O \left ( \frac{\log ...
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139 views

Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
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233 views

Weak Dirichlet's theorem for powers of primes

Let $p$ be a prime, and let $m$ be an integer coprime to $p$. Then fix a natural number $k>0$. Is there any result that is simpler than the full Dirichlet's theorem that proves the existence of ...
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Taxicab numbers.

I think most people know these numbers. Find $x,\ y,\ z,\ w$ such that $x^3 + y^3 = z^3 + w^3$ and $x,\ y,\ z,\ w$ are not equal to each other. The first is $1729$. I'm trying to figure out if ...
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$2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$

I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
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82 views

Composite $n$ such that $\sigma(n) \equiv n+1 \pmod{\phi(n)}$

I'm looking for composite $n$ such that $$\sigma(n)\equiv n+1\pmod{\varphi(n)}$$ Are there only finitely many? Can this be proved? This is Sloane's A070037 but there's not much information in the ...
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Polynomial equations in $p$ and $q$ with $p,q$ primes

Is there a non zero polynomial $R \in \mathbb{Z}[X,Y]$ such that there exists an infinite number of pair $(p,q)$ with $p$ and $q$ primes, $p \neq q$ and $R(p,q)=0$ ? I know the curve must be of genus ...
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397 views

Ramanujan Series

I have the following question involving the series of $1/\pi^3$: Can we find such expansions by using the one for $1/\pi^3$ with $1/\pi^4$ or $1/\pi^n$ etc? Note that $$\frac{1}{32}\sum_{n=0}^\infty ...
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159 views

Proof of infinitude of primes whose reversal in base 10 is also prime

Is there any proof of infinitude of http://oeis.org/A007500 primes. If you want to generate them here is trivial and naive python program. ...
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419 views

Sum of odd prime and odd semiprime as sum of two odd primes?

How to prove that each sum of odd prime and odd semiprime can be written as sum of two odd primes $(p_1+p_2p_3=p_4+p_5)$ ? Since we know that each prime number greater than $3$ is of the form $6k\pm ...
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170 views

Solutions to equation in matrix form

Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by ...
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385 views

Is this a relation between the Riemann zeta function and the Prime zeta function?

I am partly repeating myself here again. Is this a correct relation between the Riemann zeta function and the Prime zeta function? $$ \zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$ ...
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139 views

Shintani cone zeta function

Is there a procedure/algorithm for calculating sums of the form $$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$ where $$ L_i(n_1,\ldots, n_r) ...
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184 views

A subtle relationship from class field theory

Recently, I consider a problem: Let $E/F$ is a Galois extension of number field, denote the idele group of $E$ (resp $F$) by $I_E$ (resp $I_F$). There is a homomorphism induced by norm map $N_{E/F}$: ...
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A sequence avoiding 3-term power progressions

Rankin1 studied sequences of integers that avoid 3-term geometric progressions, $(a, a c, a c^2)$, e.g., $$\{1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, \ldots \} \;$$ So, $18$ is excluded ...
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Finding first n so nth fibonacci is c modulo p

This is a question I stumbled upon in an online programming contest archive. The problem statement is simple, given $c \equiv F(n)$ mod $P$ and $P$, where $P$ is a prime of form 5$k$ + 1 or 5$k$ - 1, ...
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An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
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The copy-problem : Does any block of digits appear at least twice?

Suppose, $N$ random digits have been generated. Let $X$ be the largest natural number with the following property : There are natural numbers $i$ and $j$ with $i+X-1<j$ , such that the digits $i$ ...
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68 views

Show that for a given $s$ there are a finite number of Fibonacci number of form $n^2+s$

It is well known that the last Fibonacci number $F_k$ such that $\exists \ n \in \Bbb{N} : F_k = n^2$ is $144$. Thus there are only $4$ perfect squares among the Fibonacci sequence (assuming you ...
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$\sum_{n=1}^{\infty}\frac{a_n}{10^n}$ where $\{a_n\}_{n=1}^{\infty}$ is a sequence in the ten digits {0, 1, 2 , 3 , 4 , 5 , 6 , 7 , 8 ,9}

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence in the ten digits {0, 1, 2 , 3 , 4 , 5 , 6 , 7 , 8 ,9} And consider the sum $\sum_{n=1}^{\infty}\frac{a_n}{10^n}$ $\in$ $[0,1]$ What characteristics of ...
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Zeta zeros standard normal distribution

Below is a partially scaled plot of $\vartheta (\gamma_n) - \pi (n - 3/2) ,$ where $\gamma_n$ is the imaginary part of the $n$th zeta zero, and $\vartheta $ is the Riemann-Siegel theta function, for ...
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Squarefree products of a class of primes

Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4. Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up ...
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Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of ...
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Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?

Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$): $$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) ...
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102 views

Solve $(x+1)^n-x^n=p^m$ in positive integers

Solve in positive integers: $$(x+1)^n-x^n=p^m$$ $p$ is prime, $n\ge 2$. Seemingly Zsigmondy's Theorem and LTE won't work here. Though you can tell (as suggested by user barto), using ...
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Exploiting a crypto backdoor based on a polynomial

At a capture-the-flag competition during the weekend, there was a task that involved the following polynomial over the field $F = \mathbb{F}_P$ of integers modulo $P = 571787215471557516425591$ (yes, ...
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53 views

A question on the Pell equation $x^2-pqy^2 = -1$, with prime $p,q$.

We know that a necessary but not sufficient condition such that, $$x^2-dy^2 = -1\tag1$$ is solvable is that $d$ is not divisible by a prime of form $4m+3$. It is not sufficient because the prime ...
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Question about exponentiation

Consider exponentiation on the natural numbers. Suppose we have an equation involving only the variable $x$, where there are $m$ $x$'s on one side and $n$ $x$'s on another side where $m$ and $n$ are ...
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The canonical height of a point on an elliptic curve

I am struggling with exercise 3.3 in Silverman-Tate Rational Points on Elliptic Curves. Here is the paraphrased problem with necessary background: Let $C:y^2 = x^3 + a x + b$ be a nonsingular cubic ...
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Find integer numbers $x,y>1$ such that $x|3^y+1$ and $y|3^x+1$

Find all integer numbers $x,y>1$ such that $x|3^y+1$ and $y|3^x+1$. I tried to use $\text{ord}$ but still no clue.
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Characterization of Extended Lucky Numbers

The Lucky Numbers is a sieve where one starts out with the positive integers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ... and then eliminate ...
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Is there a formula for $\sum_{n=0}^{+\infty} q^{n^3}$?

When I studyied the representation of integers as sum of squares, I found that the most powerful tool is the Jacobi Triple Product, in fact this amazing identity allows us to find more useful ...
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52 views

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
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Not using Jacobi symbol how to prove For all positive integer $n>1$ $2^n - 1 \not | 3^n-1$?

There is a proof: if $n$ is even,then $3|2^n-1$ but $3\not|\;3^n-1$,It is correct; if $n$ is odd,suppose $2^n-1|3^n-1$,then $3^n \equiv 1(\mod 2^n-1)$,then ...
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More on primes $p=u^2+27v^2$ and roots of unity

Given, $$p=u^2+27v^2=3m+1\tag1$$ and the cubic, $$x^3+x^2-mx+N=0\tag2$$ with its constant expressed in terms of $(1)$ as, $$N = \frac{1}{27}(1-3p\pm2pu)\tag3$$ and the sign $\pm u$ chosen ...
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An infinite series that gives $f(s)=s$. How could it be explained more easily?

This question loosely builds this one. Equate the following two infinite series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{4\,(s-1)} \left(1+s+\sum _{n=1}^{\infty } \left( {\frac ...
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52 views

Prove this congruence

Let $p$ be a prime of the form $4k+3$ and $m$ an even positive integer less than $p-1$. Prove that $$1^m+2^m+\cdots+\left(\frac{p-1}{2}\right)^m \equiv ...
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$\sum_{a^2<p\leq (a+1)^2}p$ Summation of primes

$$\sum_{a^2<p\leq (a+1)^2}p$$ where p is prime. Are there some known bounds for this sum?
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Numbers Made From Concatenating Prime Factorizations

I came across the following curious problem while playing around with my calculator. Take any positive integer $n$; for this example we'll use $216$. Create a sequence as follows: Factor $n$ into ...
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When does a variety have a point over a finite field for sufficiently large primes p?

Let $X$ be an algebraic variety over the rational numbers. Suppose that $X$ has positive dimension. I would like to say that $X(\mathbb{F}_p)$ is non-empty for sufficiently large primes $p$. One idea ...
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137 views

Differences between large numbers with many factors has little factors

I apologise beforehand for the informality and lack of precision in this question but it is that way because it comes from only an intuition, nothing more than a heuristic argument. Say one has two ...
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102 views

Is it decidable whether the iterates of a polynomial map are bounded?

Let $f:\mathbb{Q}^n\to \mathbb{Q}^n$ be a polynomial map with rational coefficients. Let $p\in \mathbb{Q}^n$. Is there a known algorithm that given this data determines whether or not the iterates ...
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109 views

Diophantine inequality that comes up after Vieta Jumping Hurwitz technique

I am blaming this on Prove the equality EDITTTTT: allowing $x_1 \geq x_2$ and $x_2 \geq x_n,$ I would rather not explain what that was about and the only changes are in $n=3,4,$ already settled. ...
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115 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
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34 views

Artin-Schreier Question from Corps Locaux

I have a question from Serre's book "Corps Locaux", namely question 5a in section 2 of chapter IV. It is as follows: "Let $e_K$ be the absolute ramification index of K, and let n be a positive ...
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59 views

English translation of two papers by Polya on real zeros of Fourier transform approximation to Riemann $\xi$ function

I am looking for English translation of the following two papers by Polya: [1] G. Polya, Bemerkung über die Integraldarstellung der Riemannschen $\xi$-Funktion, Acta Math. 48(1926), 305-317; ...
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133 views

Transfinite Knuth-arrow hierarchy vs. fast-growing hierarchy

Suppose Knuth arrow notation (and hence the hyperoperation sequence) is extended to transfinite ordinal indices as follows: Let μ be a large countable ordinal such that a fundamental sequence is ...
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79 views

Chinese Remainder Theorem stuff (but more advanced, certainly derived from)

First, let $m$,$n$ be coprime. Suppose we want to find: $x\equiv y\text{ mod }m$ $x\equiv z\text{ mod }n$ As m,n are coprime $\exists a,b\in\mathbb{Z}:am+bn=1$ (GCD=1 basically) Then, for ...
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72 views

Asymptotics for Mertens function

It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as: ...
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116 views

Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$

Are there any useful restrictions to the prime factors of the number $$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$$ The two smallest are 6771419 and 72153167 , which I found by trial division. The number is ...