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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177 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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180 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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56 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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123 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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64 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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99 views
Is there any new improvement in the proof or disproof of the twin prime conjucture?
I think this is not the first question about twin primes here, but my own is the latest one!
I am a postgraduate student in Mathematics interested in the field of number theory. While searching on ...
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39 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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129 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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69 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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114 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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62 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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99 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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70 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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111 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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35 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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76 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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118 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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49 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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134 views
(Not) Surprising Result on Natural Numbers as Sum of $k$-Almost Primes
I started with the following idea:
Let $P_k$ be the infinte set of all $k$-almost primes.
The counting function for $k$-almost primes less than $x$, is
$\displaystyle \pi_k(x)\sim\frac{x}{\log ...
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41 views
Composite n such that sigma(n) = n+1 mod 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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26 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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89 views
A good introduction to S unit equations
I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper.
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116 views
units in discrete valuation rings
Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
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237 views
Sylvester's Theorem and Schur Theorem
I'll probably end up asking more programming questions on StackExchange forums than math questions, but I'll lead off with a math question.
In my Number Theory class this past semester, I worked on a ...
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108 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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146 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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99 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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128 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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117 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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129 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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128 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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192 views
N(p) number of solution to x^x =1 (mod p) Miklós Schweitzer 2010
Let $p$ a prime number and $N(p)$ the number of solution to $x^x \equiv 1$ (mod $p$) in $1\leq x \leq p$ .
Prove that for sufficiently large $p$ there exist a constant $c < \frac{1}{2}$ such that
...
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68 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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108 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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655 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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184 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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55 views
Can Fermats descent 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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44 views
Exact statement of Artin reciprocity for ray classes
It seems like this particular theorem is always stated in a way that's slightly hard to interpret. Let $S$ be some finite set of primes containing all the primes of $K$ ramifying in $L/K$. Then the ...
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163 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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98 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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99 views
Feasibility of a cryptography transformation
This is a follow-up of the question: Transformation
We are given
$$g^{1/(x+m)},$$
(it is not possible to find $\frac{1}{x+m}$ due to the Discrete log problem), can we find a $k$ such that
...
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284 views
Counting Lattice Points in an $n$-sphere and Sphere Packing
Does knowing the exact number of $\mathbb{Z}^{n}$-lattice points in an $n$-sphere of arbitrary radius help one compute $n$-sphere packing density and kissing number?
Thanks!
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71 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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104 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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487 views
Solving Diffie–Hellman problem for low primitive root
What's a good way of solving the Diffie–Hellman problem when those exchanging the message have chosen a low primitive root $g$ (e.g. $g=3$)?
Of course you could brute force it but I'm interested in ...
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16 views
Computability of division of large numbers
What is the largest computable mathematical division in terms of the number of digits that can be handled by a typical desktop computer using the best available big number libraries, assuming input is ...
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25 views
The generating function for Bernoulli polynomials
The generating function for Bernoulli polynomials is given by:
$$\frac{ue^{ux}}{e^u-1}=\sum_{n\geq 0}B_n(x)\frac{u^n}{n!}$$
Now, I have the following expression:
...
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48 views
Matching numbers by $f(x)=\frac{1}{x}$
Let $0<x \leq 1$, We define a function such that $f(x)=y=\frac{1}{x}$ which results $y \geq 1$ . We have infinitely many numbers between $0$ and $1$, so we can match any $x$ to a number $y$ greater ...
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45 views
Amount of Background Needed for Number Theory Research
How much background is needed to do research pure number theory? I mean things like descriptions under 18.785 and 18.786 in http://student.mit.edu/catalog/m18b.html. I get the impression that it takes ...
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19 views
The definition of Pisot number
A Pisot number is an algebraic integer $>1 $ and all of whose conjugates have modulus $<1$. and I want to ask that is the polynomial is unique? I mean that the degree of the polynomial is ...
