Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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How to partition $S$ in this way?

Assume: $$ P =\{p_1,p_2,\cdots,p_K\}\subset \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N} $$ and, $$ f(l) = \sum_{i=1}^K \sum_{j=1}^K x_i x_j ...
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Constructing pairs of units $(x,y)$ which solve $x^2 + y^2 \equiv -1 \pmod{N}$

A classic result on the way to the Lagrange Four Squares theorem — for instance proven by Theorem 87 of Hardy & Wright, as noted by this remark on the Four Squares theorem — is that ...
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180 views

How Ramanujan find this formula

I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...
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196 views

Ramanujan and sum of four cubes

This is more a question on History than proof itself. About a decade ago, a college professor and a Math coach told us about this beautiful theorem: Every multiple of 6 can be written as a sum of ...
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59 views

Matrix representation of field automorphism

Let $K$ be the degree $n$ field extension of the field $k$, and let $\alpha_1,\dots,\alpha_n\in K$ be a basis of $K$ over $k$ (as a vector space). I read somewhere that the following matrix $$ M= ...
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234 views

Maximum length of sequence of non-coprimes of $N$ - least upper bound for Jacobsthal's function

I am looking at the length of the longest sequences of adjacent integers that are not coprime to $N$ for very large $N$. Let $F_N$ be the set of integers less than $N$ which are not coprime with $N$: ...
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121 views

Best upper bound on the number of divisors of $n$ that are larger than $N$.

I am looking for the best upper bound on $$\sum_{\substack{d | n\\ d \geq N}} 1.$$ I know that $$ d(n) = \sum_{\substack{d | n}} 1 \leq e^{O(\frac{\log n}{\log \log n})}. $$ For my application, I ...
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154 views

Each new member from the second divides the sum of all previous

All integers from $1$ to $13$ are recorded in a sequence such that each number (from the second onwards) divides the sum of all previous numbers. What numbers can be in the third place and why? I ...
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205 views

The divergence of the series of reciprocals of primes (proof check):

I just wanted to check my attempt at a proof for the divergence of: $$\sum_{n=1}^{\infty} \frac{1}{p_n} \tag{ $\star$ }$$ We begin with assuming that $(\star)$ converges. If $(\star)$ ...
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319 views

Natural extension of the divisor function $\sigma$ to the complex integers $\mathbb{Z}[i]$

The divisor function $\sigma$ of a positive integer $n$ is defined as the sum of the positive divisors of $n$. It turns out that this definition is equivalent to ...
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123 views

Proof that $t^8+2t^6+4t^4+t^2+1$ is reducible in $\mathbb{F}_p$

Prove that the polynomial $t^8+2t^6+4t^4+t^2+1$ is reducible in $\Bbb F_p$, for all $p\in \Bbb P$. Here are some examples: $t^8+2t^6+4t^4+t^2+1=(1 + t + t^4)^2\pmod{2}$ $t^8+2t^6+4t^4+t^2+1=(1 + t) (2 ...
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136 views

Primes of the form $\dfrac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
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226 views

Estimation for logarithm of Riemann zeta function

Let $\sigma >1-\dfrac{c}{2\log(|t|+3)},|t|>7/8,$ where $c$ is constant from Theorem about region without zeros of Riemann zeta function. Using the fact that $$\log \zeta(s) - \log \zeta(s_1) = ...
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158 views

Fastest Primality test using $N-1$ Factorization?

If $N-1$ could be factored easily with several small prime factors, then what is the fastest way to check $N$ for primality? Updated I'm aware of Pocklington primility test which is not good for ...
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165 views

Carmichael number factoring

The task I'm faced with is to implement a poly-time algorithm that finds a nontrivial factor of a Carmichael number. Many resources on the web state that this is easy, however without further ...
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133 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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Statement about Woodal primes.

A Woodal number is an integer of the form $n 2^{n}-1$. A Woodal prime is an integer that is both a prime and a Woodal number. Let $p$ be a prime of the form 1 mod 4. Then $p 2^{p} -1$ is never a ( ...
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208 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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279 views

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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135 views

Density of products of a certain set of primes

I have an infinite set S of prime numbers with relative density 0 (that is, $\lim_ns_n/p_n=\infty$ with $S=\{s_1,s_2,\ldots\}$ and $s_1 < s_2< \cdots$). I would like to find the size (in some ...
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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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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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370 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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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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199 views

Sums of Dirichlet-Characters over prime numbers (part 2)

This is kind of related to my previous question that was poorly stated because of misreading my own notes that I have taken on the papers I am currently reading, so no surprise that it eventually ...
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167 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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357 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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131 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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172 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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Arithmetic Functions: Evaluate $ \sigma(210)$ and $d(63)$

Evaluate $ \sigma(210)$ and $d(63)$ I'm not sure if I got this correct, so here is my attempt. By Theorem 6.3, suppose we have $n=p_1^{\alpha 1}...p_s^{\alpha s}$, then $d(n) =(\alpha_1 ...
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$\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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113 views

Will this algorithm stop before time?

For every $n \in \mathbb N$, let's define $a_0 = 0$, $$\begin{cases} a_{i+1} = 2a_i + 1 \pmod {2^n}, &\text{if it never appeared before} \\ a_{i+1} = 2a_i \pmod {2^n},& ...
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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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Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
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194 views

Prove modular equivalence related to carmichael function.

Statement if $n=p_1p_2p_3\cdots p_r$ (prime numbers are distinct) and $2 \le a \le n-1$ then prove that $$a^{k\lambda} \equiv \left(1+\sum_{p_i\mid\gcd(a,n)}\left(\dfrac{n}{p_i} \right) ...
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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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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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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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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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Ramification index of infinite primes

I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an ...
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69 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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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 ...
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55 views

Closest Pair between 2 sets of powers

Assume $a$ and $b$ are natural numbers, $A=\{a,a^2,a^3,\cdots\}$ and $B=\{b,b^2,b^3,\cdots\}$, find $\min\left|a_i-b_j\right|$ where $a_i\in A$ and $b_j\in B$. For example, if $a=3$, $b=10$, then ...
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187 views

start studying advanced topics in number theory

I am a first year undergrad and have had elementary course in Number Theory which includes only basic introductory topics like: divisibility,gcd-lcm,primes,congruences, number theoretic functions etc. ...
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93 views

How find this equation all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$ My try: If $n>1$ is odd, then note $$\left(n^{\frac{n+1}{2}}\right)^2<n^{n+1}+n-1<\left(n^{\frac{n+1}{2}}+1\right)^2$$ so ...
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90 views

Algorithm for comparing the size of extremely large numbers

Is there a simple algorithm to decide which of the numbers $$a \uparrow ^b c \text{ and } d \uparrow ^e f$$ is the bigger one ? Using the hyperoperation, the numbers can be denoted with ...
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50 views

Automorphism group of an L-function

I define the notion of Galois class of L-functions as follows: $A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true: 1) $A$ is a subset of the ...
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48 views

List of Primes in UFD

Are there websites/databases containing lists ordered by norm of prime/irreducible elements in domains like $\mathbb{Z}[\sqrt{-2}]$ and $\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$ for easy ...