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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13 views

Character sums over witt rings of finite length

please I want to know if there is somme references for the studying the characters sums on the group of the Witt vectors of finite length
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14 views

Adding or multiplying by a constant each time to a set of numbers to obtain desired numbers

Suppose there is a finite set of integers $X = \{x_1,x_2,..,x_n\}$. For each single arithmetic operation, one can only operate on every integer in the set. (For example, we cannot multiply one number ...
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30 views

What is the probability of the sum of elements in $\mathbb{Z}_{N^2}^*$ to be multiplicatively inverted?

Given a set of elements $ x_i \in \mathbb{Z}_{N^2}^*$ how can we express the probability of $\sum{(x_i)}^{-1}$ to have a multiplicative inverse $\mod \mathbb{Z}_{N^2}^*$, , where $N=pq$ for two safe ...
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18 views

The estimation of $\sum^{K_0+K}_{k=K_0+1} \min\left\{ U, \frac{1}{\left< \alpha k + \beta \right>} \right\}$

I have some difficulty with understanding the proof of the following theorem: Suppose $\alpha$ is a real number which has the form $\alpha = \frac{h}{q} + \frac{\theta}{q^2}$, $(q,h)=1$, $q \geq ...
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12 views

On t-core partitions

How exactly can one define what is known as a t-core partition? I know (vaguely) that it involves the definition of what is known as "Hook numbers". Anyone cares to provide a link or explain it?
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42 views

For which $p$ is a number a square in $\mathbb{Q}_p$?

I have some numbers $r \in \{-1, 2, \frac{4}{5}, \ldots\}$ and have to find those primes $p$ for which $r$ is a square in $\mathbb{Q}_p$, i. e. is a solution of the equation $X^2 = r$ in ...
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27 views

Simple number theory inequality

$$-b < -r \leq 0\text{ and } 0 \leq r' < b \implies -b < r'-r < b$$ how is that implication possible? I'm going over the proof for the division theorem mainly the uniqueness part ...
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57 views

If $N$ divides $a$ and $N$ divides $b$ then

If $N$ divides $a$ and $N$ divides $b$ then $N$ divides $a \cdot b$? Is the statement true? I mean. $$a \equiv 0 \pmod{N}$$ $$b \equiv 0 \pmod{N}$$ $$\implies ab \equiv 0 \pmod{N}$$ But I am ...
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22 views

Are the expressions (a/b) %c and (a%(b*c)) /b equivalent?

As far as I know, to determine (a/b)%c, we need to determine (b^-1)%c which can be done using extended euclid, fermat's theorem, euler's theorem or there may be some other way, but what we must need ...
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25 views

Looking for reference giving exposition on Pell's Equation

I know the continued fraction convergence method of finding a solution of Pell's equation . I am looking for other methods of finding the smallest or at least one solution of Pell's equation which is ...
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22 views

Writing payoff matrix and understanding zero-sum game

I have an example in my book that states the following problem: i) Write the payoff matrix of the two players for rock-paper-scissors, where the losing player gives $1 to the winning player. ...
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35 views

How to determine if $3$ is a square $\mod 97$

The answer is that it is since $10^2 \equiv 3 \mod 97$, but how do we determine that $3$ is in fact a square without having to find explicitly the square that is $3 \mod 97$ ($100$ in this case)?
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23 views

Exponential and regular Diophantines?

I am looking for a reference on connections between exponential and "regular" (polynomial) Diophantine equations. For example, I was wondering about the Catalan-Mihailescu problem and I thought of the ...
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25 views

How can I verify the result of modular exponentiation

I ask a computer to calculate $x^y \pmod z$, where $x,y,z$ are all large numbers. How can I verify the correctness of the result returned by the computer. I assume that I myself cannot afford ...
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126 views

Some feedback on the sequence $ a_n=a_{n-1}+4\phi_n $

This sequence is present at OEIS A171503. There, Jacob Siehler explains how this sequence correspond to the number of matrices with determinant one and how this number grows as we allow to vary in ...
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35 views

Existence of a Fermat Liar

I'm trying to prove that at least one Fermat liar exists for a composite number n when $gcd(\phi(n), n-1) > 1$. I can see how if n was prime, then the gcd would equal 1, but I'm not sure how to ...
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14 views

Weighted sums with floors

I am interested in the following problem. Suppose $\alpha_1, \cdots, \alpha_n$ are positive real numbers, each at least as large as $1$. Let $D$ be a large positive integer (with respect to the ...
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50 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...
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13 views

Relative primes and exponent function

Let $k \neq 0$ be an integer, $\psi(n)$ denote number of $r \leq n$ relative prime to $n$ and $$f_n(r)=e^{2\pi i k\frac{r}{n}}$$Prove that: $$\left|\frac{\sum_{(r,n)=1}f_n(r)}{\psi(n)}\right| \to 0$$ ...
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38 views

Elementary solution to the Mordell equation $y^2=x^3+9$?

I've recently been wondering how to solve the equation of mordell for k=9, namely: (y^2=x^3+9). It reduced to solving the Thue equation (|a^2-2b^3|=3).Interestingly, the equation has several ...
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30 views

Characteristic polynomial and prime splitting.

In studying prime splitting in non-abelian extensions, we study Artin l-functions,wherein the Euler factor is defined to equal the characteristic polynomial of the matrix representing the Galois ...
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19 views

Sum of divisors of n/n = t + 0.5

How can i find the sum of all abundant numbers less than 10^18 which follow the property (sigma(n)/n) = t + 0.5 where sigma(n) gives the sum of divisors of n , t is a positive integer. In other ...
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22 views

Minimize product of distinct numbers in set with no 3-term geometric progression

I would liked to find a set, $S$, of $n$ natural numbers where there exists no 3-term geometric progressions (i.e. $ab = c^2$, $a, b \in S$ and $c \not\in S$) that minimizes the number of distinct ...
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32 views

Fibonacci polynomial summation

The $n^{th}$ value of a polynomial ($S_n$) of order $k$ is a polynomial in $x$ is given by : $\left(\frac{S_n}{x^n}\right)$= $\sum_{j=0}^n \left(\frac{{F_j}^k}{x^j}\right)$ Where ${F_j}$ is the ...
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47 views

Prove that $ax^2 + by^2 \equiv c \ ( \mod{p})$ has integer solutions

Let $p$ be a prime number and $a, b, c$ integers such that $a$ and $b$ are not divisible by $p$. Prove that $ax^2 + by^2 \equiv c \ ( \mod{p})$ has integer solutions Well, this problem can be ...
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22 views

Use of inclusion-exclusion principle

I have N numbers.My question is that how many ways I can take 4 numbers such that their GCD(greatest common divisor) is 1. Can this problem be solved using principle of inclusion-exclusion. If so how ...
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36 views

Can you find squares in this class?

For a problem I am working over, I would like to prove that numbers of the type are not squares $p(l^4+6l^2m^2-3m^4)$ where $p,l,m$ are integers an $p$ prime. I have already found various necessary ...
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37 views

Bernoulli Numbers -Identity?

I have been searching for an identity that would help me simplify an equation. Let, $B_m(x)$ be the Bernoulli Polynomial. What are min and maximum bounds on $B_m(0)$? (essentially the last term in ...
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19 views

Compute or determine an upper bounde for a problem in probability

Suppose $M$ is a even number. compute the exact probability or determine the upper bound for the below probability: $Pr\{M-p=q\thinspace$ | $q$ is prime $\}$.($p$ is prime and arbitrary and $p ...
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33 views

Finding Explicit Function from a reclusive formula

I have been working on a project that will move much faster if I can write a recursive formula as an explicit formula. Let, $f(m-1,i)=i*f(m,i)-f(m,i+1)$ $i \in \mathbb{N}$ Thus, ...
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14 views

Which transformations preserve this?

Let $a,b,x,y,z\in \mathbb{Z}$ (with $a,b$ given) and consider the equation $a(x^2+y^2+z^2)=b(xy+yz+zx)$. Consider transformations taking $x$ to $px+p'y+p''z$, $y$ to $qx+q'y+q''z$ and $z$ to ...
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9 views

Invariant subspaces of Bishop Operator

Let $\Omega=\{\alpha\in(0,1)|\exists$ a sequence of mutual prime integer pairs $(\mathit{p_{n},q_{n}})$ such that |$\alpha$-$\frac{\mathit{p_{n}}}{\mathit{q_{n}}}|<$ ...
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45 views

Infinite product involving prime numbers

In an article I am reading the author states that the product over all prime numbers $p$ $$\Pi_p(1-p^{-p}),$$ is about $.722$. Why? I believe this should be related to the Riemann zeta function.
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31 views

How to prove this task?

We have the plane grid, and in every field, we write a natural number. In all of the fields, the numbers written down next to them have a fix average. Prove, that all numbers must be equal. I ...
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13 views

is the degree of an L-function a semiring homomorphism?

Consider the biggest subset of the intersection of the Selberg class and the set of automorphic L-functions closed under product and tensor product (that corresponds on the automorphic side to ...
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20 views

Define binary relation $pRq$ on the set of positive odd primes. $ x^2\equiv p \mod q$.

Define the binary relation $p R q$ on the set of positive, odd primes to mean that there is an integer $x$ satisfying $0 <= x < q$ such that $x^2 \equiv p \mod q$ holds. Is $R$ an equivalence ...
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34 views

Finite matrix power over $\Bbb Z$

$p=\text{prime}$. $p[\Bbb N_{T_1\leq T_2}]=\{0\}\cup \{p^t:t\in\Bbb Z, T_1\leq t\leq T_2\}$. Given $T\in\{0\}\cup\Bbb N$, what is largest $s\in\Bbb N$ such that there is a partition $$0=T_0\leq ...
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23 views

Problem with re-writing a parametrization in the Maple software

I tried run the first parametrization for equation $$aX^2+bXY+cY^2=jZ^2$$ in this page Solutions to $ax^2 + by^2 = cz^2$ with Maple software. But it didn't work. Where is my mistake? My Maple code ...
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16 views

Bound on $|\delta|$ when $\alpha\beta=\gamma+\pi\delta$

Suppose $\alpha,\beta,\gamma,\delta,\pi$ are Gaussian integers and $\pi$ a prime element of norm $>2$. If $|\alpha|,|\beta|,|\gamma|<|\pi|$ and $\alpha\beta=\gamma+\pi\delta$, is it true that ...
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45 views

Count good numbers that are multiple of 7 upto M

Given a number of $N$ digits with $A[1]$ denoting the first digit , $A[2]$ second digit and $A[n]$ last digit from left to right. It is called good number if ...
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13 views

Scheme theoretic definition of field extensions

You can think about a number field $K$ as the spectrum of its ring of integers. Is there anything equivalent for a field extension?
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24 views

How to find integer solution for bilinear transformation?

Let y = (ax + b)/(cx + d), where a, b, c, d integer constants, is there any technique to find integer solution of x & y?
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34 views

Characterization of two sets

I am interested in the following problem: Let $n \in \mathbb{N} $ . We define the function $S_n: \mathbb{Z_n^*} \to \mathbb{Z_n} $ \begin{align} S_n(\bar a) := \bar 1 + \bar a + \bar a^2 + ...+ ...
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26 views

Parametrization to $4ax^2+4by^2=3t^2$

fix $a$ and $b$ as rational numbers. If $x,y$ and $t$ be variable then how can I have a parametrization solution to this equation? $$4ax^2+4by^2=3t^2$$
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29 views

Question about procyclic groups

Say I have a local field $K$, complete for a discrete valuation. Let $I^t$ be the tame inertia group and let $p > 0$ be the residual characteristic. I know that $I^t \cong \widehat{\mathbf{Z}}'(1) ...
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28 views

Gauss circle problem reference request

I'm dealing with the Gauss circle problem. I have heard that for the number of lattice points this formula is valid: $N=\sum\limits_{n=0}^{[r^{2}]} 4 \cdot \prod\limits_{p \equiv 1 mod 4} ({1+k(p)})$ ...
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45 views

Fermat primes question

The regular n-gon is constrcutibe by ruler and compass when the odd prime factors of n are distinct fermat primes. How do I go about proving the case for φ(n) is a power of 2? A helpful hint or ...
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31 views

What's a Schwartz-Bruhat Function

Let $X$ be a locally compact abelian group and $f: X \rightarrow \mathbb{C}$ a continuous map. There are several definitions of what it means for $f$ to be a Schwartz-Bruhat function. If $X = ...
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35 views

The product of two integer sequences with bounded gaps

Given that two non-negative integer sequences $(a_i)_{i\in \mathbb{N}} , (b_i)_{i\in \mathbb{N}}$ such that $a_{i+1}-a_{i}$ and $b_{i+1}-b_{i}$ take only finite values. define $c_i=min(a_i,b_i)$ Is ...
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25 views

definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...