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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279 views

Ramanujan's sums

Are the series expansions of arithmetic functions in terms of Ramanujan sums computationally useful? I didn't think they would be, but they seem to be good approximations even when summed with few ...
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128 views

Find total number of sets of integers which satisfy a given equality and inequality

Compute the total number of different sets of integers a1, a2,..,an which satisfy the following equality and constraints: $$ ...
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216 views

Möbius sums and Eulers totient function

Let $\phi(a)$ be Euler's totient function, and $\mu(k)$ be the Möbius function, how can I prove that for all $a$, $$\phi(a)a=\sum_{\substack{\gcd(a,r)=1,\\1\leq r\leq ...
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30 views

Question about the definition of weekly modular of weight k

I'm reading the book A First Course In Modular Forms and it defines the term weakly modular of weight $k$ as following: Let $K$ be an integer. A meromorphic function $f:H\rightarrow\mathbb{C}$ is ...
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171 views

unique factorization of matrices

If I have a set of matrices, call this set U, how can I make this a UFD (unique factorization domain)? In other words, given any matrix $X \in U$, I would be able to factorize X as $X_1 X_2 ... X_n$ ...
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146 views

Number of solutions to polynomials over finite fields

Let $m = p_{1}p_{2}\cdots p_{s}$. Let $N_{f}(n)$ denote the number of solutions to $f \equiv 0 \pmod{n}$. If $f = f(x)$, a polynomial in the single variable $x$, then from the Chinese Remainder ...
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147 views

solution count of quadratic form congruences

Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
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27 views

group law on weil-chatelet group

Is there a reference for the gemetric definition of the group law on the Weil-Châtelet group of an Abelian variety more recent than the original Weil's paper ("On algebraic groups and homogeneous ...
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129 views

Elementary divisors of a finite abelian group

Suppose $A$ is a finite abelian group. (a) Extract from the function $h_A(n) = |\{x \in A : x^n = e\}|$ ($n \in \mathbb{Z}$) the elementary divisors of A using the fact that for a cyclic group $C$ of ...
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55 views

absolute galois group of Puiseux series with coefficients in $\bar{\mathbb{F}_p}$

Let $K$ be the field of Puiseux series with coefficients in $\bar{\mathbb{F}_p}$ (the algebraic closure of the field with elements). What is the absolute Galois group of $K$ ? Thank you to anyone ...
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223 views

Struggle proving maximal ideals principal in $\mathbb Z[\varphi]$

I was worried that my proof isn't right so I want to know if there are any mistakes in this and if this way can work? Thank you very much. We want to show every maximal ideal $\mathfrak m$ of ...
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78 views

Primes clasification

For all numbers $N > n$ ( $n$ is positive number), let $p$ be an odd prime $<$ $(2N)^{1/2}$ and $d = 2N-2p+1$, then there exist at least an odd number $d$ which does not contain any odd prime ...
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159 views

Question about the elementary divisors of a special matrix

I have the following question: Is there a closed formula for the elementary divisors of the Matrix $M={(m_{ij})}_{i=1,...,n,\ j=1,...,k}$, where ${m}_{ij}$ is the greates common divisor of $i$ and ...
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27 views

Series function help

I want to find a function such that $$ \sum_{0<j<n/k } f(kj)=1 $$ Where the sum j is taken over the natural numbers, And the series is satisfied for all integers k and n, I was thinking of ...
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37 views

Does an irreducible polynomial in K(t)[x] give an irreducible polynomial in K[t][x]

Let $K[t]$ be the ring of polynomials over a field $K$. Let $K(t)$ be its fraction field. Let $f$ be an irreducible polynomial in $K(t)[x]$. There exists an element $a\in K[t] $ such that $af$ is in ...
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94 views

Proving a simple inequality

Can someone show that the inequality bellow holds? $$ f(n) \leq f(n+1) \ $$ Where $$ \frac{\sum\limits_{k=1}^n \Lambda(k) {k}/{n}\lceil{n}/{k}\rceil{}\{ n/k \}}{\sum\limits_{k=1}^n \Lambda(k)}=f(n)$$ ...
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29 views

Dilation mod p, small gaps

If I have a set $A \subset \{1,2,3,...,\epsilon p \} \subset \mathbf{Z}/p \mathbf{Z}$ does there exist a dilation $ \lambda $ such that $ \lambda A$ has no gap larger than $s$ (where $\epsilon = ...
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113 views

Permutations within a specific boundary

Let's have the following sequence of natural numbers: 1, 2, 3, 4, 5, 6, 7, 8. The permutations of these 8 numbers are equal to 8!. We can obtain some of these permutations by adding and subtracting ...
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60 views

Inequality help

Can someone help me prove the inequality, $$ \frac{\sum\limits_{k=1}^n \Lambda(k) \frac{k}{n}\lceil\frac{n}{k}\rceil\{ \frac{n}{k} \}}{\sum\limits_{k=1}^n \Lambda(k)}<\ \frac{\sum\limits_{k=1}^n ...
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51 views

How good might $|\vartheta(x)-x|$ be?

This question is about Chebyshev's first function, $\vartheta(x) = \sum_{p\leq x}\log p.$ Assuming the truth of the Riemann hypothesis, $|\vartheta(x) -x|= O(x^{1/2+\epsilon})$ for $\epsilon > 0.$ ...
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54 views

Find the orders..

$\newcommand{\ord}{\operatorname{ord}}$ Find the orders below: \begin{align} & (a) \quad \ord_{11}5 \\ & (b) \quad \ord_{7}4 \\ & (c) \quad \ord_{23}22! \end{align} For the most part, I ...
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86 views

$\int_{\gamma}\frac{dz}{\sqrt{1-z^2}}=2\pi$ Along the path $\gamma(t)=2e^{it}$ for $0\leq t\leq 2\pi$ Implies Sine is $2\pi$ periodic

Ok so to back up a bit, by a trig substitution we have for $f:(-1,1)\rightarrow\mathbb{R}$: $$f(x)=\int_0^x\frac{dt}{\sqrt{1-t^2}} = \arcsin(x)$$ Now according to the notes here: Elliptic Functions ...
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163 views

Matiyasevich polynomial proof

Can someone provide a proof, or a link to a proof, of why does the Matiyasevich polynomial always generate primes for the nonnegative results? Any help will be appreciated.
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49 views

Rationality and convergences

I observed that convergence of partial fractions and I am writing this samll story for seeking more clarity and justifications etc. If f(n) = $\frac{p(n)}{q(n)}$, where p and q are polynomials of n ...
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67 views

the 2-rank of field

Let the field $K=\mathbb{Q}(\sqrt{p_1}, \sqrt{p_2 q}, i)$ where $p_1, p_2 \equiv 1 \mod{4}$ and $q \equiv 3 \mod{4}$, kronecker(2,$p_1$)=1 and kronecker(2,$p_2$)=kronecker($p_1$,$p_2$) = ...
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638 views

Legendre and Jacobi symbols

I have a problem with Tonelli-Shanks algorithm with numbers $n = 87463$ and $p = 17$. Solutions are supposed to be $x_1 = 7$, $x_2 = 10$, but I get $11$ and $6$. First with sieving I get a list of ...
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298 views

The Lucas Theorem and facts

I have studied the Lucas theorem and I encountered the following facts. How to deduce the following facts from The Lucas theorem? (1) If d, q > 1 are integers such that , $$\binom{nd}{md}$$ ...
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61 views

series: can the result be zero for a continuous interval of its argument?

I'm considering the series $$ f_c(x) = \sum_{k=c}^\infty \left( c^{k-1} \binom{k}{c} \cdot \prod_{j=1}^{k-1} (x-1/j) \right) $$ where the parameter $c \in \mathbb N ,c \gt 0$ and fixed for a certain ...
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199 views

Lucas' theorem Consequence

Lucas' theorem consequence $$\binom {m}{n}=\ \prod_{i=0}^k\;\ \binom {m_i}{n_i} \pmod{p},$$ $$m=m_k\;p^k+m_{k-1}\;p^{k-1}+\cdots +m_1\; p+m_0,$$ $$n=n_k\;p^k+n_{k-1}\;p^{k-1}+\cdots +n_1\;p+n_0$$ ...
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96 views

Infinitesimals and infinite elements among the transseries

In the quest for extensions of $\mathbb{R}$ and $\mathbb{C}$ that contains infinitesimals, infinities (and even more exotic beasts like $\omega - 1$ and $\sqrt{\omega}$) I came across the theory of ...
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48 views

A question regarding the method followed in Cohen & Selfridge's paper on covering systems.

Note: I have posted this question on MO before. No one replied, so I am reposting it here. I am reading this paper by Cohen and Selfridge that deals with covering systems. Its link is ...
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138 views

A special factorization

Suppose that monic polynomial $f(x)\in\Bbb Z[x]$ such that for all $m\in\Bbb Z$, $m>1$, there's no integers $\langle r,r_1,\ldots,r_m\rangle$ such that $f(r)=f(r_1)\cdots f(r_m)$. Is there any ...
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85 views

Is there any way to find summation of first n perfect numbers?

I was studying properties of perfect numbers when this question clicked me. Is there any way to find summation of first n perfect numbers? Is the only way to sum them up is to write them down? Sorry ...
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132 views

diophantine equation with squares over 3 variables

I am trying to find solutions for this diophantine equation $$x^2+y^2+x^2y^2=4z^2$$ I am looking for advice on a procedure to find all positive integer solutions for this equations.
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171 views

How to find the last non-zero digit in ${^n\!P_k} $?

What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
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89 views

Applications and motivation of η-quotient generators and algorithms

I previously asked this question on mathoverflow but got no satisfactory answers so I'm posting it here as well: So initially Dummit, Kisilevsky and McKay found all Dedekind $\eta$-products which are ...
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151 views

Four squares theorem etc

We have studied Lagrange's four-square theorem and is denoted by g (2) = 4. i.e., any number can be expressible in sum of squares of four positive integers. Now my question is, here g (2) = 4, where 4 ...
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94 views

How to calculate $L'(1,\chi)/L(1,\chi)$ in SAGE?

Question as in title, where $L(s,\chi)$ is the Dirichlet $L$-function associated with the nontrivial character modulo $3$. Please provide complete SAGE code. Thank you in advance.
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145 views

explicit formula for norm map of Kummer extensions

Since it is particularly easy to write down a basis of a Kummer extension $K=k(\mu)/k$ (where $\mu^n=a \in k$) as a $k$-vector space, I suspect that it is should not be terribly hard to write an ...
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46 views

Minimal modulus for the finite field NTT

I need your support. Suppose I am performing an NTT in a finite field $GF(p)$. I assume it contains the needed primitive root of unity. I am using it to compute the convolution of two vectors of ...
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82 views

Binary forms of degree n

Im trying to show that the binary form $x^{n−1}+x^{n−2}yα+x^{n−3}y^2α^2+...+y^{n−1}α^{n−1}$ is bounded below by $ c*y^{n−1}$ where c is some explicit constant. For the case n=3 this is fairly easy, ...
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132 views

expectation of vector

Let vector $c\in 2N $ is such that first $m$ of its coordinates are $1$ and the rest are $0$ ($c=(1,\ldots, 1,0, \ldots, 0)$). Let $\pi$ be $k$-th permutation of set $\{1, \ldots, 2N\}$. Define ...
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102 views

Formula simplification, finding maximum

It happens that I have to use combinatorial and I am trying to study by my own this interesting subject. I have two questions on which I am stuck. Any help will be appreciated. Let $n_i\in \{0, 1, ...
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76 views

Finding the random $r$ in a Paillier encrypted message with knowledge of the private key.

In the Paillier cryptosystem, suppose that I know a Ciphertext encrypted with some unknown random $r$ i.e. $$C = (g^m r^n) \bmod n^2 $$ I know $g, n$, the prime factorization of $n$, i.e., $pq$. I ...
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32 views

Number of energies of the free Laplacian.

Given the Selberg trace formula, and the fact that the eigenvalues of the operator $\Delta -1/4 =T$ are the zeros of the Selberg zeta function, then would it be correct to say the number of ...
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120 views

Basic example of extensions of residue fields.

Can anyone think of a simple example of the following: $B/A$ is an integral extension of DVRs with quotient fields $L$ and $K$ and residue fields $\bar{L}$ and $\bar{K}$, $L/K$ is finite dimensional ...
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272 views

variants of geometric series

My question can $\displaystyle \mathbf{\sum_{n \geq 0} a^{\lfloor n \sqrt{2}\rfloor}}$ be expressed as the sum of rational functions in a? Here $\lfloor \alpha \rfloor$ is the floor function, the ...
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152 views

Efficient factorion search in arbitrary base

A factorion in base $N$ is a natural number equal to the sum of the factorials of its digits in base $N$. So, the decimal factorions are: $1 = 1!$ $2 = 2!$ $145 = 1! + 4! + 5!$ $40585 = 4! + 0! + 5! ...
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160 views

Forcing and divisibility

I'm going to bring together a couple of seemingly unrelated questions that I've asked here. This may be silly. Or maybe not? Imagine that $n$ is some sort of infinitely large integer, and thus so ...
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132 views

Binary sequences in primes

Is anything known about these problems? If we make a string S of 0's and 1's with 1 in n'th position if the the nth prime $p_n$ is of the form $1+m 2^{9^{9^{9^{9}}}}$, else 0, does every finite string ...