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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151 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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159 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 ...
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316 views

Easiest way to prove that a subset of even integers is closed under multiplication?

What's the easiest way of showing that; $2\mathbb{Z}\setminus (4n-2)\mathbb{Z}$ is closed under multiplication? (I'm trying to show that $(4n-2)$ is a prime element of $2\mathbb{Z}$ by showing ...
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184 views

Galois Group over Ring of Integers

Suppose we have a quadratic (Galois) extension of $\mathbb{Q}$, call it $k$ with Galois group $G$. If we look at the ring of integers inside of $k$, call it $\mathcal{O}_k$, is it true that ...
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95 views

Are the signs of these eigenvalues from this Hermitian matrix equal to the Möbius function?

I am partly repeating myself here. Are the signs of these eigenvalues from this Hermitian matrix "c" equal to the Möbius function? Eigen99 in the Mathematica code is the list of eigenvalues for a ...
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117 views

Does number theory have any role in the proof of convergence of Fourier series for certain functions?

Does number theory have any role in the proof of convergence of Fourier series for certain functions? I vaguely remember reading in a book on signal processing, way back, that the proof (original ...
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1k views

Teach me a simple, efficient division algorithm

I want to implement arbitrary-precision arithmetic in JavaScript for non-negative integer numbers. Long division isn't efficient if instead of the usual 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) there ...
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215 views

Dirichlet's Class Number and its connections with the $GL(2)$

i posted the same question on MO,but cant get an answer so i am trying here note:all those who answer my question just mention the question number in their reply so that i can tally them,thanks a ...
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70 views

Can Fermat's descent for $x^4+y^4=z^2$ be interpreted on a conic?

Fermat proved the Diophantine equation $$(x^2)^2 + (y^2)^2 = z^2$$ has only solutions $(0,0,0)$, $(0,\pm 1,\pm 1)$ and $(\pm 1,0,\pm 1)$ using "infinite descent". The conic $C : X^2 + Y^2 - 1$ has a ...
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228 views

$(a^n +b^n)/((ab)^{n-1}+1)$ is a perfect $n^{th}$ power

Let $a,b$ be positive integers satisfying $$(ab)^{n-1}+1 \mid a^n +b^n.$$ Then how to show that the number $\frac{a^n +b^n}{(ab)^{n-1}+1}$ is a perfect $n^{th}$ power of an integer? Another question: ...
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116 views

Potential computational questions that could be asked about p-adic numbers and Galois Theory

I have an exam on P-Adic integers (and a bit on Galois Theory) that my professor said would be very computational, but he never does any examples of the theorems he proves in class. He said the exam ...
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80 views

Are limits on exponents in moduli possible, if the modulus is relatively prime?

I asked a similar question to this recently. Here, I consider an arbitrary, but fixed, modulus m, which is relatively prime to x and y. Can anybody extend the answer given in the previous question? ...
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112 views

Is that series-transformation known in the context of divergent summation

Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the ...
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5 views

Proving $\gcd(N^a-1,N^b-1)=N^{\gcd(a,b)}-1$.

I have come by one solution only, but things were derived too quickly without me understanding how or why. How does knowing that $\gcd(a,b)$ is a factor and a and b, actually derive that ...
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10 views

Roadmap to $p$-adic numbers: where a self-learner should look for references

TL;DR at the end of the question. I’m currently trying to learn as much as possible about p-adic numbers. I’m not sure what is the most fascinating part of the theory, but the use of the adjective ...
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10 views

Does there exists a positive $t$ that satisfy this given condition?

I am curious about the validity of my claim concerning the equations: $(2k-1)t+1$ (1) $(2k^2-2k)t+(2k-1)$ (2) where $k=2,3,4,...$ My claim is for almost all $k$ or for infinitely many $k$, there ...
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21 views

a question in De koninck and luca's analytic number theory

what is your idea,can you introduce a book for these kind of problems?
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12 views

Is there a match between this modified prime pi function and the Log integral function?

Table T is defined as through the properties that accumulated row sums give prime numbers, while accumulated column sums give composite numbers. ...
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4 views

Space complexity of the GNFS

The time complexity of the general number field sieve is generally quoted as $L_n\left[\frac{1}{3}, \sqrt[3]{\frac{64}{9}}\right]$. I'm looking for a credible reference (something that can be quoted ...
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28 views

Least pair of numbers having at least $k$ distinct prime factors

Consecutive numbers with less than $k$ prime factors? shows that for every $k$, there is a pair $(n/n+1)$, such that $n$ and $n+1$ both have at least $k$ distinct prime factors. The object is to ...
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19 views

Primality radius and quadratic reciprocity law

Given an integer $n>1$, I say that $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are primes. Goldbach's conjecture asserts that every integer greater than $1$ admits a primality radius. ...
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44 views

Number theory / decimal representation

Prove that for any $n\in\mathbb{N}$ there exists a number $m\in\mathbb{N}$ such that the decimal representation of $m^2$ has $n$ ones at the beginning and some combination of $n$ ones and twos at ...
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18 views

Selberg combinatorial identity

I am reading Granville's article on bounded prime gaps and in Section 4.5, he says that suppose $L(d)$ and $Y(r)$ are sequences of numbers supported only on the square-free integers. If $$Y(r) := ...
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36 views

Find all pairs of positive integers $(x,y)$ : $x(x+1) = y(y+1)(y+2)$

Find all pairs of positive integers $(x,y)$ : $$x(x+1) = y(y+1)(y+2)$$ I was able to find only two pairs: $(2,1)$ and $(14,5)$ and looks like no more exists. How to prove it?
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18 views

Reduced Residue class problem

I need to Prove that when $j \ge 3$, then every reduced residue class modulo 2j may be written in the form $((−1)^a)(5^b)$ , where a = 0 or 1 and $1 \le b \le 2^{j−2}$, and in which the integers a and ...
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14 views

Difference between consecutive squarefree (cubefree) numbers

The jumping champions for the greatest difference between consecutive squarefree numbers are : ...
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15 views

A Greatest Common Divisor Question

What is $GCD(a_0a_1\bmod N,a_0a_2\bmod N)$ where $GCD(a_0,a_1)$, $GCD(a_0,a_2)$, $GCD(a_1,a_2)$, $GCD(a_0,a_1,a_2)$ could each be non-trivial? ($a\bmod N$ here is remainder of $a$ divided by $N$).
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30 views

Factorization of the sine

I am working on the Basel problem for a project for my Mathematics study. I need to proof that one could write the sine as a factorization of its linear roots. I know the proofs is in general done bye ...
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15 views

The asymptotic upper density of $\{xy \colon 1\le x\le y\le 2x\}$

Find the asymptotic upper density of the set $\{xy\, \colon\, 1\le x\le y\le 2x\}$. In other words, let $S$ be the set of all integers which can be expressed as $xy$, for some positive integers ...
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46 views

Solve an equation of the prime counting function

The problem is, Find all the positive integral values of $x$ for which we have, $$\pi(p_n-x)=\pi(p_{n+1}-x-1)$$where $\pi(x)$ denotes the number of primes not exceeding $x$. I don't know where ...
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13 views

Are integral combination and linear combination the one and the same in the field of number theory?

My question is a trivial question as to the exactness of the meaning of integral and linear in the study of the number theory. Do they hold the same meaning?
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14 views

What are the elements with the minimal absolute values in cyclotomic extensions of the integer ring?

We can easily see 1 has the (nonzero) minimal absolute value in the integer ring. In the Gaussian integer ring and the Eisenstein integer ring, 1 is also the minimal (complex) absolute value. However, ...
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17 views

Is there a multi-perfect number which is NOT of the form $2^n(2^{n+1}-1)u$?

The even perfect numbers have the form $2^n(2^{n+1}-1)$, where $2^{n+1}-1$ is a (Mersenne-)prime. Is every multi-perfect number of the form $2^n(2^{n+1}-1)u$, where $u$ is some odd number ? It is ...
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8 views

number of distinct numbers of the form $e(k^2(4a)^{-1})$

Let $q$ be a large prime. Define $e(n)=\exp\{2\pi i\frac{n}{q}\}$. What is the cardinality of the set $\{e(k^2(4a)^{-1}): a,k\in\mathbb{N}\}$? Here $a^{-1}$ means the multiplicative inverse of $a$ in ...
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61 views

Singularities in the weighted projective space

Is there an explicit criterion for checking that a hypersurface $f=0$ of degree $d$ and in $\mathbb{P}(a_0,\ldots,a_n)$ is smooth ? I could not convince myself that the criterion $\nabla f\neq 0$ ...
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75 views

How to show that the equation $7^3d^2-3^3c^2=1$ has infinitely many integer solutions ?

How to show that the equation $7^3d^2-3^3c^2=1$ has infinitely many integer solutions ? Please help . Thanks in advance
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24 views

Could you please explain the algorithm for the below given number series generated by Excel Fill series?

Below images represent the number series that are obtained using the Excel Fill Series If you input 13, 16, 17 then 19.33333, 21.33333, 23.33333, 25.33333,.. is generated. If you input 34, 424, ...
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17 views

Prove that $2$ is a quadratic residue modulo primes of the form $7.2^n-1$, where $n\geq 3$

Prove that $2$ is a quadratic residue modulo primes of the form $7.2^n-1$, where $n\geq 3$ I have no attack on this. I know that this works for $n=5$. Could someone give me a hint?
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30 views

Ideal class groups of a real quadratic field.

I am trying to compute the ideal class group of a real quadratic field of an integer such that it is congruent to $1\pmod 4$ and $1\pmod 8$ and it's Minkowski bound is around 4. The problem is that I ...
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26 views

Primality test similar to Pocklington

I'm working on a series of number theory proofs and I'm stumped on this one. The idea is to extend each result for the subsequent proof. I just succeeded in proving the following: Suppose $n > 1$. ...
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23 views

Sums of two squares theorem

Theorem: For each $n\in\mathbb{N}$, the number of integral solutions $x,y$ to the equation $n = x^2 + y^2$ is given by $4\sum_{d|n} \chi_1 (d)$, where $\chi_1$ is the Dirichlet character: $$ ...
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36 views

Provide me notes on Riemann zeta function to boast my knowledge to use in Research on Analytical Number Theory

I need your help. I want to study the Riemann zeta function from the very basic level, its concepts, theorems, solved problems etc. I am assigned one problem from Analytical Number Theory related to ...
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35 views

what are the benefits of a factorial number system?

After reading an article about factorial number system. It tells that you can present any number in a factorial system and in if you have a number $a_{n-1}...a_2a_1a_0$ in factorial number system, you ...
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6 views

Unique symmetric multilinear form associated to a form

Let $F(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a form of degree $d$. In an article I am reading, it says to associate to $F$ the unique symmetric multilinear form $F(\mathbf{x}_1| ... | ...
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28 views

Largest known multi-perfect number (excluding perfect numbers)

What is the largest known multi-perfect number (excluding the perfect numbers) ? [2, 94; 3, 32; 5, 9; 7, 11; 11, 2; 13, 8; 17, 1; 19, 5; 23, 1; 29, 2; 31, 2; 37, 1; 43, 1; 53, 1; 59, 1; 61, 2; 67, ...
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29 views

Number of representations of sums of four squares?

I was told that multinomial expansion can be used to determine how many representations of four squares a number like 53 has? I have a number theory textbook and have done some googleing neither has ...
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13 views

Torelli Shanks Algorithm - Repeated Squarring Method

This algorithm is using when you want to find a square root of a number in a given moduli. I can't see the idea behind this algorithm, so can someone explain it in a simple way?
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18 views

How many kind of basis function to approximate an arbitrary function

I am finding a list algorithm to approximate an arbitrary function. Such as Bernstein, he said that a linear combination of Bernstein basis polynomials $$B_n(x) = \sum_{\nu=0}^{n} \beta_{\nu} ...
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20 views

Finding the m-th coefficient of an equation

Does anyone know how to find the m-th coefficient of $-t=(3+2\sum_{k=1}^{\infty} \frac{t^{2k}}{(2k)!})(\sum_{n=0}^{\infty} D_n \frac{t^n}{n!})$? The answer is: $0=3 \frac{D_m}{m!}+2\sum_{k=1}^{m/2} ...