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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-2
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1answer
50 views

Find minimum possible area of brush

A rectangular brush has been moved right and down on the painting. Consider the painting as a $n × m$ rectangular grid. At the beginning an $x × y$ rectangular brush is placed somewhere in the frame, ...
2
votes
1answer
46 views

smallest element order $p$ in $\mathbb Z^*_{p^2}$

I would like to write an efficient algorithm to find the smallest element of order $p$ in $\mathbb Z^*_{p^2}$, where $p$ is a prime number. Therefore I calculate $a^{p-1} \pmod{p^2}$ for every ...
0
votes
2answers
52 views

Solving a Linear Diophantine Equation

A Linear Diophantine Equation is of the following form: $Ax+By+C=0$, where $x_1 \leq x \leq x_2$ and $y_1 \leq y ...
0
votes
2answers
20 views

Summing powers of complex root of unities gives 0

I have a question regarding a proof. Let $z_N$ denote the complex N'th root of unity, from which we have the identities $(z_N)^n=1$ $\sum_{i=0}^{N-1}{(z_N)^i}=0$ Now let $N=r\cdot t$ and let ...
0
votes
0answers
6 views

What is the relation between the upper bound,low bound of simple continued fraction expansion of quadratic algebraic numbers and the integer

What is the relation between the upper bound,low bound of simple continued fraction expansion of quadratic algebraic numbers and the integer As we know,$\sqrt{14}= [3;1,2,1,6,1,2,1,6…]$,let $B_u,B_l$ ...
5
votes
1answer
72 views

When is the next palindrome?

Okay, this is more just for fun than anything else. I'm driving in my car today, (true story) and my odometer is about to hit $81,818$. So, being a math nerd and all, I immediately see the pattern ...
1
vote
0answers
10 views

Does a random binary sequence almost always have a finite number of prime prefixes?

Does a random binary sequence almost always have a finite number of prime prefixes? Specifically, let $x = \sum_{1 \le i}{2^{-i} \cdot x_i}$ with $x_i \in \{0,1\}$ be a random real in $[0,1)$, $X_i = ...
3
votes
1answer
44 views

Some questions on the formation of the BSD conjecture

I'm quite curious how Birch and Swinnerton-Dyer formed their famous conjecture in the beginning of 1960s. I read some paper of Birch and Swinnerton-Dyer, as well as some paper of Tate and several ...
4
votes
1answer
98 views

Existence of $\{a_{n}\} $ and $\{b_{n}\}$ such that $a_{n}(a_{n}+1)|(b^2_{n}+1)$

Show that: there exist two sequences $\{a_{n}\}$ and $\{b_{n}\}$ that are monotonically increasing (or $a_{n+1}>a_{n},b_{n+1}>b_{n},\forall n\in N^{+}$) and for any positive integer $n$ ...
1
vote
2answers
39 views

A stricter Fermat's little theorem

By Fermat's little theorem we know that $a^{p-1} \equiv 1 \pmod{p}$ for all primes p. But it is often possible to find $x$ such that $a^{x} \equiv 1 \pmod{p}$ and x < p - 1. Is there anyway to ...
0
votes
1answer
51 views

Can we not apply the Hensel Lifting Lemma in this case?

Check if the equation $x^2=-1 \text{ in } \mathbb{Z}_2$ has a solution, and if it has, calculate the three first positions of the solution. So, we are looking for a solution $\pmod 2$, one solution ...
0
votes
1answer
41 views

PNT and Maximal Prime Gap Connection? [closed]

Are the formulas for estimating maximal prime gaps related to the Prime Number Theorem? Why do they both use the natural logarithm? Is the formula in the conjecture that there is always a prime ...
0
votes
2answers
70 views

Prove that if $p$ is a prime and $k$ is an integer, there are two integers $x$ and $y$ that satisfy $x^{2} + y^{2} + k \equiv p$ [closed]

Prove that if $p$ is a prime and $k$ is an integer, there are two integers $x$ and $y$ that satisfy $$ x^2 + y^2 + k \equiv 0 \pmod p. $$
0
votes
1answer
80 views

Distance between powers of 2 and 3

As we know $3^1-2^1 = 1$ and of course $3^2-2^3 = 1$. The question is that whether set $$ \{\ (m,n)\in \mathbb{N}\quad |\quad |3^m-2^n| = 1 \} $$ is finite or infinite.
2
votes
2answers
59 views

Chinese Remainder Theorem RSA

I want to solve the following modular quadratic equation: $x^2 \equiv 188 \pmod {437}$ using the fact that $437$ can be factorized by the primes as: $19⋅23$. So far I have done: $$x^2 \equiv 188 ...
3
votes
2answers
109 views

The Conjecture That There Is Always a Prime Between $n$ and $n+C\log^2n$

Let $n$ be 113. Use $n+C\log^2n$ to find the next consecutive prime or at least approximately how far away it is. Will you show me how to work this out step by step to show me how to use this formula? ...
1
vote
3answers
55 views

Finding if a number is prime by looking at the sum of their digits

Take a number $N = \overline{abcdef...}$ where $a, b, c, d,e,\dots$ are the digits of $N$. Let $k$ be the sum of those digits : $a+b+c+d+e+... = k$ If $k$ is any of ${1, 2, 4, 5, 7, 8 }$ then $N$ ...
2
votes
1answer
54 views

When is “being a linear algebraic $k$-group” preserved?

Let $G$ be a linear algebraic group over a field $k$, with Char$(k)=0$. What "group-theoretical operations" preserve the property of "being a $k$-linear algebraic group"? For example When ...
3
votes
1answer
53 views

Number of answers of equation amongs odd natural numbers

How many answer The following Equation has, in set of odd natural numbers? $x_1+x_2+...+x_k=n$, $k \equiv^2 n$ Solution: Comb ( [(n+k)/2]-1, k-1), comb means combination. how we get this?
1
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2answers
46 views
1
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0answers
62 views

IMO 1983 Solution - Day 1 Problem 3

The questions goes as follows: Let $a$ , $b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc - ab - bc - ca$ is the largest integer which cannot ...
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votes
0answers
71 views

The curve with irreducible and has genus 2

Sketch the curve $f(x,y) = x^2 -x + y- y^5$, and prove that it is irreducible and has genus $2$. And, how to conclude that, $f(x,y)$ has only finitely many rational points as well as only finitely ...
1
vote
1answer
39 views

How to compute this probability: uniform distribution of two random variables

Let $p$ be a prime number. Let $i,j \in \{1, \dotsc, p-1\}$ be fixed numbers. Let $A$ and $B$ be two random variables, where $A \in_{u.a.r} \{1,...,p-1\}$ and $B \in_{u.a.r} \{0,...,p-1\}$. ...
0
votes
1answer
59 views

Prove all multiples of $U$ contain all the digits $0$ to $9$

I have to prove that the number $U = 5263157894736842101$ is a "constant number" (that is, every positive multiple of this number contains all the digits from $0$ to $9$ at least one time). In ...
1
vote
1answer
34 views

Number of different vectors.

Let's say that I have a vector with 6 elements. I put two wedges in the vector, i.e., at position 2 and position 6, for instance. And when I say put a wedge, it means... for every time you traverse ...
0
votes
2answers
45 views

If $c | ab$, then $c | a$ or$ c | b$

I need help proving/disproving the implication, If $c | ab$, then $c | a$ or $c | b$ So far, I got Assume $c | ab$ then $ab= cl$ for some integer $l$ Now what should my next step be?
4
votes
1answer
60 views

A number theory puzzle

I recently came across the following number theoretic puzzle. Suppose I've infinitely many cards, each with a natural number written on it. Given any $n\in \mathbb N$, the number of cards which have a ...
1
vote
0answers
81 views

Why is $2^{16} = 65536$ the only power of $2$ less than $2^{31000}$ that doesn't contain the digits $1$, $2$, $4$ or $8$ in its decimal representation

$65536$ is the only power of $2$ less than $2^{31000}$ that does not contain the digits $1$, $2$, $4$ or $8$ in its decimal representation. http://en.wikipedia.org/wiki/65536_%28number%29
2
votes
1answer
61 views

Closed form for $\sum_1^\infty 1/p^n$

I was wondering if there are some studies on closed forms for the sum $$\sum_{p \in \mathbb{P}}^\infty \frac{1}{p^n},$$ where $\mathbb{P}$ denotes the set of prime numbers. Obviously I know that ...
2
votes
3answers
34 views

Congruence between binomial coefficient and integer part.

If $p$ is a prime number , prove that $\forall n \in \mathbb{N}, n\geq p:$ $$\binom{n}{p} \equiv \Bigg[\frac{n}{p}\Bigg] (\text{mod }p)$$ where [ ] is the integer part i´been trying this problem ...
0
votes
2answers
50 views

How to show the congruence involving the divisor function

Prove that if $n \in \mathbb{N}$; $n \equiv -1$ $(mod 24)$ $\Longrightarrow $$ \sigma(n) \equiv 0$ $ (mod 24) $ where $\sigma $ is the divisor function. my try: if $n \equiv -1$ $(mod 24)$ ...
0
votes
2answers
35 views

The Greatest Prime Less Than $n$

Let $n$ be any natural number greater than 2. Let $l$ be the greatest prime less than $n$. When $n$=3, $l$=2. When $n$=10, $l$=7. When $n$=25, $l$=23. What is the relationship between $n$ and $l$? ...
1
vote
1answer
35 views

Showing $\mathbb{Z}+\mathbb{Z}(\frac{1+\sqrt m}{2})$ is a Euclidean domain

Does anyone know an elementary proof for the following proposition? It is stated without proof in my textbook: Let $m$ be a negative squarefree ineteger with $m = 1 \pmod 4$. Then the integral domain ...
2
votes
3answers
49 views

How solve system of congruences?? [closed]

Can anyone help me? $$\left\{\begin{array}{l} 100x - 99y \equiv 2 \pmod{210} \\ 97x + 98y \equiv 3 \pmod{210} \end{array}\right.. $$
0
votes
2answers
18 views

Is proof by modular arithmetic appropriate in this syntax?

I have a question which asks: Prove there are no integer solutions for the equation: $$4x = y^2 +1 $$ To prove, lets take $\pmod4$ of both sides, such that: $$ 4x\pmod4 = (y^2 +1)\pmod4$$ $$ ...
0
votes
2answers
49 views

Twin Prime Constant

How would one prove that the twin prime constant $$C_2 = \prod_{p > 2}1-\frac{1}{(p-1)^2} > 0$$ Simply computing the product for a large number of terms isn't rigorous, and simply establishes ...
0
votes
1answer
43 views

Exact Equivalence of Legendre's Conjecture Impossible?

If the upper bound for the prime gap above $n$ is such that $n$+4$\sqrt{n-1}$$\geq p$, where $n$ is any given natural number and $p$ is the next prime after $n$, then Legendre's conjecture is true. If ...
24
votes
1answer
518 views

Prove that both $x+y$ and $xy$ are rational, under some conditions

As a result of the answer I got for this question - Irrational solutions to some equations in two variables - I was wondering if the next statement is always true: Let $x,y$ be real, irrational ...
0
votes
0answers
25 views

Completed group ring

Let $p$ be a prime number. Consider $\mathbb{Z}_p$ the ring of $p$-adic integers. If $G$ is a profinite group, define $\mathbb{Z}_p[[G]]$ to be the inverse limit $\lim\mathbb{Z}_p [G/U]$ where $u$ ...
2
votes
0answers
27 views

Solutions $\pmod {p^n}$

We have a solution $x_0 \pmod 7$ of $x^2 \equiv a \pmod 7$. To find a solution $\pmod {7^2}$, we are looking for $x_1 \in \mathbb{Z}$ such that $$x_1^2 \equiv 2 \pmod {7^2} \text{ such that } x_1 ...
1
vote
5answers
78 views

How did we find the solution?

In my lecture notes, I read that "We know that $$x^2 \equiv 2 \pmod {7^3}$$ has as solution $$x \equiv 108 \pmod {7^3}$$" How did we find this solution? Any help would be appreciated!
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votes
2answers
91 views

lowest denominator to lie between to rational numbers.

What's the lowest $ m\in \mathbb {N} $ such that the exists an $ n $ with $1/3 >\frac {n}{m}>33/100$? note that there used to be a typo in the inequality which gives the opposite sign I'm on my ...
0
votes
0answers
19 views

Need help at the multiplication

The sum and the multiplication in $\mathbb{Z}_p$ correspond to the the sum and the multiplication of powerseries. Example for sum: $$(3 \cdot 7^0+4 \cdot 7^1+2 \cdot 7^2)+(5 \cdot 7^0+3 \cdot 7^1)= ...
1
vote
1answer
51 views

Make whole array as zero

Given an array of N elements some of which are positive and some are negative now some positive valued elements can give their value to negative valued elements.Now we need to make whole array as zero ...
2
votes
2answers
42 views

When is a sum of consecutive roots of unity an integer

Let $\xi \neq 1$ be an $n$th root of unity. When is a sum of the form $$ 1+\xi+\xi^2+\ldots+\xi^r, \quad 1 \leq r \leq n-1, $$ an integer? What are the possible integers? I suspect that the answers ...
1
vote
0answers
45 views

Can one prove the divergence of $\sum \frac{1}{p}$ by the absolute convergence criterion of infinite products?

Euler proved this celebrated theorem that $\sum \frac{1}{p}=\infty$ by using the product formula that $\displaystyle \zeta (s)=\prod \left( \frac{1}{1-p^{-s}}\right)$. Now I thought of another ...
2
votes
1answer
45 views

Divisibility Property

I am trying to justify the following result: Let $p,q$ be integers such that $GCD(p,q) = 1$. Then for all $n \in \mathbb{N}$ exists an integer $j_n$ such that $q^{j_n}t = t \ (mod \ p^{2n+1}), \ ...
8
votes
6answers
810 views

Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational?

Let $x$ be rational with $0<x<1$ and let $y$ be the rational defined by $y = 1 - x.$ Let $n$ be any natural number with $n>2.$ Then I want to prove that $$x^{(1-1/n)}+ y^{(1-1/n)}$$ will ...
1
vote
1answer
51 views

How to write a general proof to prove that for all $m$, $m^n \geq n^m$

After proving $m^n \geq n^m$ for several values of $m$, it can be inferred that for every $m$ there's a $k$ such that if $n \geq k$, $m^n \geq n^m$. In other words, this can be generalized as: For ...
2
votes
0answers
35 views

Group of Units in Cyclotomic Integers

I'm trying to show that for any $p$-th root of unity $\zeta$, where $p$ is an odd prime, we have $\mathbb{Z}[\zeta]^{\ast} = (\zeta)\mathbb{Z}[\zeta + \zeta^{-1}]^{\ast}$. Obviously the $(\zeta)$ ...