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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188 views

All the small primes close together yet again

$$ \begin{align} 2254 & = 2\cdot7\cdot7\cdot23 \\ 2255 & = 5\cdot11\cdot41 \\ 2256 & = 2\cdot2\cdot2\cdot2\cdot3\cdot47 \\ 2257 & = 37\cdot61 \\ 2258 & = 2\cdot1129 \\ 2259 & = ...
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117 views

How many solutions to $x^3+y^3 = z^3\pm 1$ for $z$ less than a bound?

Assume $a,b,c, N$ as positive integers, let primitive be $\gcd(a,b,c) = 1$ and, $$a^2+b^2 = c^2\tag{1}$$ Supposing you want to know how many solutions there are with $c$ less than a bound $N$. ...
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31 views

Closest bounding function for lower minimal Goldbach p's peaks

Mathematica Input: Let: GoldbachP[n_?EvenQ] /; n > 3 := Block[{m = PrimePi[n/2], p}, While[! PrimeQ[n - (p = Prime[m])], m--]; p] valueset[n_] := Module[{M}, M = {k = n; With[{kmin = ...
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63 views

Euler's conjecture about ${a_1}^n+{a_2}^n+\cdots+{a_{n-1}}^n=b^n$

I've known the followings: Euler's conjecture : There is no non-trivial integer solution $(a_1, \cdots, a_n, b)$ such that $${a_1}^n+{a_2}^n+\cdots+{a_{n-1}}^n=b^n$$ for $n\ge 3.$ It is known that ...
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59 views

Does there exist an operation which partitions any fraction into the sum of the minimum number of unit fractions?

Motivation : I've been interested in finding an operation which partitions a fraction into unit fractions. The following is one of the operations which I've found. Let's start a rational number $q_0$ ...
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102 views

Decryption of an Encrypted Message

Suppose we are given sending a message to two people: A and C. A and C have the same RSA encryption modulas: R=(some arbitrary number, say) 454564515456465465465156. But A and C have two different ...
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56 views

Problem in Diophantine Equation

It seems from experiments $\frac{n^2+n}{2}=2^a+3^b$ has finite number of solutions $(n,a,b)$. What is the trick to prove these type of result?
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174 views

All Sufficiently Large Squares, Represented as Sum of Two Semiprimes

Define a semiprime to be the product of two (not necessarily distinct) primes, $p_iq_i$. Conjecture: All squares $\ge 4^2$ are representable as the sum of two distinct semiprimes. Case 1: Squares ...
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122 views

Dirichlet's Theorem

Dirichlet's theorem on arithmetic progressions states that for any two positive integers a and b, if gcd(a,b) = 1 then the arithmetic progression $t(x)=ax+b$ $(x ≥ 0)$ contains infinitely many prime ...
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79 views

Parametric Equation solving over integers

I have a question on my mind I am trying to solve. However I am stuck at a point. If you could help, I would be very pleased. $$\frac {x_2y_2z_2}{x_2y_2+y_2z_2+x_2z_2}=\frac ...
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116 views

Mathematical areas that are applied to Rubiks Cube solution

I had seen about Group Theory being applied to rubik's cube and infact the solution algorithms are also based on group theory... I want to know whether other mathematical fields like "optimization" or ...
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54 views

Smallest $r$ such that $\sum_{k=0,\,\, i+kr = qm}^{\lfloor (n-i)/r \rfloor} \binom{n}{i + kr} = 0 \pmod n$

I want to find the smallest positive integer $r$ such that $$\sum_{k=0,\,\, i+kr = qm}^{\lfloor (n-i)/r \rfloor} \binom{n}{i + kr} = 0 \pmod n$$ where $n=pq$, and every $i+kr = qm$ for some $m$ is ...
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105 views

A Gauss sum over a field.

Let $K$ be a field (not necessarily $\mathbb C$) and let $\zeta=\zeta_n$ be a primitive $n$th root of unity in $\bar K$. I would like to know if there is a formula calculating $$ \sum_{k=1}^n ...
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25 views

(Reference Request) Canonical forms for Real and Complex binary forms of low degree.

I am asking for a reference for Canonical forms for Real (and Complex) binary forms of low degree with respect to the natural action of the Real (and Complex) special linear group $SL_{n}(\mathbb{R})$ ...
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72 views

For squarefree $i$ what is $\sum_1^{n} \frac{1}{i}$?

For squarefree $i$ what is $\sum_1^{n} \frac{1}{i}$ ? I use $\sum_{\sqrt{n}>m>1} \mu(m) ln(\frac{n}{m^2}+\frac{1}{2})$. I know about the connection with $\zeta(2)$ and ...
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119 views

Gelfond-Schneider Constant $2^{\sqrt{2}}$

Someone knows a proof (books , articles) that $2^{\sqrt{2}}$ is irrational ? Without using that $2^{\sqrt{2}}$ is transcendent. Any hints would be appreciated.
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52 views

Algebraic curve over rings $\mathbb{Z}/n\mathbb{Z}$

Can you give a reference about algebraic curve over rings $\mathbb{Z}/n\mathbb{Z}$? I'm very interested in analogy Hasse-Weil theorem, R-R theorem... Thank you.
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625 views

What are real life applications of Diophantine equations?

Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.
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261 views

Twist of elliptic curve

It is continuation of this question: explict form of the equation of elliptic curve Let $p$ is prime and $p = 3 ($mod $4)$. $q = p^n$. It is easy to see that $E: y^2 = x^3 + x$ has $1 \pm 2q + q^2$ ...
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172 views

Can an odd perfect number be divisible by 101?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
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88 views

Prime Numbers and Primitive Roots

Let $p_1$, $ p_2$, $p_3$ different prime numbers. Let $N = p_1p_2p_3$. Given $(p_1-1)|(N-1), (p_2-1)|(N-1)$ and $(p_3-1)|(N-1)$, prove that for every number $a \in \Bbb N$ such that $\gcd(a,N) = 1$ ...
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106 views

If $a_{n+1}=\lfloor 1.05\times a_n\rfloor$, does there exist $N$ such that $a_N\equiv0 \ $(mod$\ $$10$)?

I've known the following theorem. Theorem: Supposing that $a_{n+1}=\lfloor 1.05\times a_n\rfloor$ for any natural number $n$, there exists $N$ such that $a_N\equiv0 \ $(mod$\ $$10$) for any integer ...
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237 views

Convergence of infinite series over prime numbers

Consider the following sum: $$\sum_{p\in\mathcal{P}}\frac{1}{p},\mathcal{P}=\{p|p\equiv1(\mod3),p\mathrm{\ is\ a \ prime \ split \ in\ } \mathbb{Q}(\sqrt[3]{3})\}$$ Question: Is this sum convergent ...
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138 views

Existence of prime pairs

Will there always exist a prime pair of the form (p, p+l) for any l where gcd(p,l) = 1 and l is even? Can we always conjecture that there exist infinitely many prime pairs of the form(p, p+l) when ...
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86 views

How can prove this $\sum_{k=0}^{p}\binom{p}{k}(\pm i\sqrt{3})^k\equiv 1\pm (i\sqrt{3})^p-p\sum_{k=1}^{p-1}\frac{(\mp i\sqrt{3})^k}{k}(mod p^2)?$

for any prime $p>3$,show that : $$p\sum_{j=0}^{p-1}\dfrac{(-3)^j}{2j+1}\equiv \left(\dfrac{p}{3}\right)(mod p^2)$$ where $\left(\dfrac{p}{3}\right)$meaning Legendre symbol. This is proof: we have ...
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101 views

fractional part of rational power arbitrary small

I think that {$a^n$} (where {x} is x (mod 1)) , where $a$ is fixed rational greater than 1 and $n$ is positive integer, is dense in $[0,1]$ is unsolved. However what about {$a^n$} is arbitrary small ...
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77 views

How to find such $f(n)$, that satisfies $f(n) \equiv x_n \pmod{ 12}$, where $x_n$ is 12-periodic sequence?

How to find such $f(n) \in \mathbb{N}, n \in \mathbb{N}$, that satisfies $f(n) \equiv g(n) \pmod{ 12}, g(n) \in \mathbb{N}$, where $g(n)$ is 12-periodic sequence? For example $n^m,n \in \mathbb{N}, ...
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62 views

number of ways of expressing a number as sum of 5 squares modulo 10

Look at the function $r_5(n)$, which is defined by the number of ordered integers $(a,b,c,d,e)$ which satisfy $a^2+b^2+c^2+d^2+e^2= n$. Now, I have conjectured that the unit's digit of $r_5(n)$ is 2 ...
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81 views

Deriving this recursive expression for Riemann Prime Counting Function?

Why does this work? $f(n,k,1)=0$ $f(n,k,j)= \frac{1}{k} - f(\lfloor\frac{n}{j}\rfloor, k+1, \lfloor\frac{n}{j}\rfloor) + f(n,k,j-1)$ Here, f(n,1,n) computes the Riemann Prime Counting Function. ...
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37 views

The number of sets of size k, containing distinct numbers $\leq m$, which sum to $n$

$$Y(n,m,k) = \left|\{s \mid s \in \mathcal P (\{1,2,3...m\}),Sum(s) =n, |s| = k \}\right|$$ Is what I'm going for. Does anyone know if this function has a name? Or if there's an algebraic formula ...
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120 views

Why these two problems lead to same answers?

Suppose these two problems: Problem 1: $$\min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {|\sum_{1\leq i\leq N_p}^{N_p}x_ie^{\frac{2\pi l}{N}p_i}| \over {\sum_{i=1}^{N_p} x_i^2}} \quad \equiv \quad ...
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260 views

Index of ideals in rings of integers in number fields

Let $R$ be the ring of integers of a number field, or more generally, a finite index subring of it, and let $P$ be a prime ideal of $R$. Is there exist a good bound for the index $[R:P^n]$ in terms of ...
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102 views

Is there a generic approach to Generating Function of periodic sequences?

Recently I read on wiki (see here): "Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones." ...
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134 views

Extensions of valuations

I'm trying to understand how a valuation $v$ of a field $K$ extends to an algebraic extension $L$. In chapter 8 of his book ANT, Neukirch first chooses a $K$-embedding $\tau : L \longrightarrow ...
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72 views

$\theta(x) = O(x)$ in the prime number theorem

In the Newman short proof of the prime number theorem (http://www.maths.dur.ac.uk/~dma0hg/prime_number_theorem_zagier.pdf) Zagier states that the fact that $2^{2n} >= e^{\theta(2n) - \theta(n)}$ ...
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65 views

Order of summation of Moebius function with summations of fractional parts as coefficients

I want to show that $\displaystyle\sum_{i=0}^n\left(\mu(i)\sum_{j=1}^{\lfloor\frac{n}{i}\rfloor}\{jx\}\right)=O(n)$ for $x\in (0,1)$. I have tried to use the result that ...
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63 views

solutions to $f(y) = n(n + 1) \ldots (n + m - 1)$

I was reading a paper about solutions to $f(y) = P(m)$, where $f(y) \in \mathbb Z[y]$ and $P(m) = n(n + 1) \ldots (n + m - 1)$ is a product of $m$ consecutive integers ...
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259 views

Finding the maximum XOR metric

I'm trying to find a way to find n keys (x bits) where the XOR distance metric between them would be greatest. By XOR distance metric I just mean the value when two keys are XORed together. So for ...
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163 views

Query on Brahmagupta-Fibonacci Identity

According to Brahmagupta-Fibonacci Identity, for $p=q\cdot r$ we can prove if any two of the integers $p,q,r$ are of the form $a^2+n\cdot b^2,$ the third must of the same form This is probably a ...
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58 views

How can I solve this using BIT?

I found a nice math problem, but I still can't solve it, I tried to find one solution using google and found that it can be solve using the Binary Indexed Tree data structure, but the solution is not ...
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57 views

Ring of the residual classes $(\Bbb Z/p\Bbb Z)^\times$? $p$-adic integer?

In a recent question we raised the theorem: for a given prime $p$ and a given power $m$ the representation of any positive integer $n\in \Bbb N$ in the form: $$ n=(a_u p - b_u) \; p^m$$ is unique ...
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47 views

A question about the solutions of a diophantine equation

I would like to know if it's possible to find the solution of the following equation: $$x^k+y^h=z^{kh}$$ in which: $$\{x,y,z\}\subset\mathbb{N}$$ given $k$ and $h$ with: $$\{k,h\}\subset\mathbb{N}$$
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124 views

Pythagorean triples and the unit circle

use the lines through the point $(1, 1)$ to describe all the rational points on the circle $x^2+y^2=3$. Why isn't this possible?
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37 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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83 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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85 views

Extending a rational entry matrix to an orthogonal matrix.

Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the ...
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34 views

How to show this space is complete?

Let $\mathbb F_q$ be the finite field of $q$ elements. We let $K_{\infty}$ to be the field of formal power series in $x^{-1}$ over $\mathbb F_q$. If $$ \alpha = \sum_{- \infty}^{t} a_i x^i, $$ then ...
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132 views

Modified Arithmetic-Geometric Mean

Let $\{x_n\}$ and $\{y_n\}$ be defined iteratively, $x_0:=\beta >1, \ y_0:= 1$ and $x_{n+1}= \frac{x_n+y_n}{2}$, $y_{n+1} = (x_n.y_n)^{\frac{1}{2}}$; i.e. they are respectively the arithmetic and ...
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47 views

Synthetic Division with mods

$x^4+x+1$/ $2x^2$+1 In $F_5$ (means mod 5) I said let the leading coefficient be 2. Since $3$ $*$ $2$ - $5$ $*$ $1$ $=$ $1$, choose 3 to multiply $3$($2x^2$ $+$ $1$)= $x^2$ + $3$ (this is ...
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163 views

Is zero a cluster point of $n\sin n$?

I have seen somewhere that if $0\le \alpha<1$, then zero is a cluster point of the sequence $n^\alpha \sin n, n=1,2,\cdots$. My question is what if $\alpha=1$? Or $\alpha>1$?