Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

learn more… | top users | synonyms (1)

0
votes
2answers
167 views

Is there an odd integer other than 1 that is the sum of its divisors? [closed]

Is there an odd integer other than 1 that is the sum of its divisors (e.g., 6=1+2+3 and 1,2,3 are the divisors of 6)?
0
votes
4answers
83 views

Finding roots of equation in $\Bbb{Z}_{14}$, $\Bbb{Z}_{17}$

Find all the roots of the polynomial $f(x) = x^2 - 9x + 6 = 0$ in $\mathbb{Z}_{14}$. Use this to factor $f(x)$ in as many ways as possible. Repeat this process for $x^2 - 9x + 3$ in ...
0
votes
2answers
46 views

The product of two numbers that can be written as the sum of two squares

Prove that if n is the product of two numbers that can be written as the sum of two squares then n can be written as the sum of two squares.
0
votes
3answers
65 views

Solving different types of Diophantine equation [closed]

In each of the following three equations I need help in finding all solutions in positive integers : i) $\dfrac 1x+\dfrac 2y-\dfrac3z=1 $ ii) $\dfrac 1{x^2}+\dfrac 2{y^2}+\dfrac 3{z^2}=\dfrac 23$ ...
0
votes
2answers
32 views

Prove that the legrende $(\frac{\frac{p-1}{2}}{p})=(-1)^{\frac{(p+5)(p-1)}{8}}$

Let $p>2$, where $p$ is prime . $$\text{Prove:} \ \ (\frac{\frac{p-1}{2}}{p})=(-1)^{\frac{(p+5)(p-1)}{8}}$$ $$\ \ (\frac{\frac{p-1}{2}}{p})(\frac{2}{p})=(\frac{p-1}{p})=(\frac{-1}{p})$$ So we ...
0
votes
5answers
54 views

How to prove that $x^2≡2(\bmod 3)$ is not a complete square

Let $m$ be the product of first n primes (n > 1) , in the following expression : $$m=2⋅3…p_n$$ I want to prove that $(m-1)$ is not a complete square. I found two ways that might prove this . My ...
0
votes
2answers
29 views

Is $n^2>n^2/m + nm$ for $m$ taking some value between $1$ and $n$.

This is not a homework question. I am working on this paper and I need to come up with a proof for this, I will not include this is in the paper but I need to be sure of this or I will end up looking ...
0
votes
2answers
60 views

Integer roots of quadratic equations

Does there exist real $b,c$ such that each of the equations $x^2+bx+c=0 \space,\space 2x^2+(b+1)x+c+1=0$ has two integer roots ?
0
votes
2answers
62 views

Divisibility of sum of powers: $\ 323\mid 20^n+16^n-3^n-1\ $ for which $n?$

I found this question in my Math Challenge II Number Theory packet: Find all positive integers $n$ that satisfy $323|20^n+16^n-3^n-1$. I don't even have any idea how to approach this question. Any ...
0
votes
3answers
128 views

At least $p^2-p$ solutions to $x^2+y^2+z^2 \equiv 1 \mod p$

I'm trying to solve a graph theory problem that relies on for any prime $p$ there being at least $p^2-p$ solutions to $x^2+y^2+z^2 \equiv 1 \mod p$. I believe its true but my number theory is rusty ...
0
votes
3answers
46 views

Number Theory help required

Hi, can someone help me solve this? $$110x \equiv 3 \pmod{73}$$ So far, I have completed the the Magic table and found the GCD. I got ...
0
votes
1answer
219 views

How come if $\ i\ $ not of the following form, then $12i + 5$ must be prime?

I know if $\ i\ $ of the following form $\ 3x^2 + (6y-3)x - y\ $ or $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$, then $\ 12i + 5\ $ must be composite number, ...
0
votes
1answer
51 views

Sum of the number of factors of natural numbers equality

I have come upon the problem of proving an equality of the divisor function How can you prove that the the expressions for the summatory divisor function are the same; $ \sum_{m=1}^n d(m) = ...
0
votes
1answer
50 views

How find this positive integer $(a,b)$ such $\left(\frac{a}{b}-\left[\frac{a}{b}\right]\right)\left[\frac{a}{b}\right]=2013$

let $a,b$ is positive integer numbers,and such $$\begin{cases} \gcd(a,b)=1\\ b\le 100\\ \left(\dfrac{a}{b}-\left[\dfrac{a}{b}\right]\right)\left[\dfrac{a}{b}\right]=2013 \end{cases}$$ Find the pairs ...
0
votes
4answers
51 views

Gcd and divisibility properties

suppose $a|c$ , $b|c$ and $gcd(a,b)=1$ .Then show that $ab|c$. Here $a,b,c$ are all real numbers.Can i start from the properties of divisibility as if $a|c$ and $c|b$ then $a|b$?
0
votes
1answer
39 views

How to get the following reductions?

I am reading a textbook on number theory and encounter the following reduction steps. But I cannot understand why are them sound. Who can give me more detailed reductions? Suppose $p_i$ are primes, ...
0
votes
3answers
70 views

How large do my $2$ primes need to be to “guarantee” a longevity of security for my RSA-encrypted plaintext?

I am currently attempting to learn RSA. Most of the literature I am using is at least a few years old, if not older. Given the advancements in computing and improvements in attacking RSA, I am wanting ...
0
votes
2answers
116 views

${p^km \choose p^k} \equiv m \pmod p$

Let $p$ be a prime number and $m, k$ two positive integers. Then ${p^km \choose p^k} \equiv m \pmod p$. I've been trying to demonstrate this lemme all the day. Have you got any suggestion? Thank you ...
0
votes
3answers
54 views

Number Theory: Remainders

“ Let $a, b \in \mathbb{Z}$ and that $0<a<b$. Given $b=qa+r$ where $0\leq r<a$. Prove that $r$ is always less than $\frac{b}{2}$. ” I have played around with several examples and have ...
0
votes
2answers
50 views

Multiply the money game

Two players A and B are playing a game. The game is as follows: the player having the turn can multiply the money with a particular number between 2 to 9 and pass the money to other player. For ...
0
votes
2answers
58 views

A question of divisibility.

Let $p$ and $q$ are relatively prime integers. Consider $S = \{\frac{p}{q}\} + \{\frac{2p}{q}\} + \{\frac{(q-1)p}{q}\}$, where $\{x\}$ is the fractional part of the real number $x$. Prove that $2S$ ...
0
votes
1answer
55 views

How can we analytically know 34! equals to zero for the following C# code?

Hopefully my question below fits to this site. I think so! Background Integer type (int) in C# takes 32 bit memory. Its value spans from ...
0
votes
2answers
30 views

Multiplicative Functions

Prove that if $f$ and $g$ are multiplicative, then so is $$F(n) = \sum_{d|n} f(d)g\left(\frac nd\right) $$ I have an understanding of how to prove it for a single function, but 2 functions and a ...
0
votes
2answers
148 views

Numbers between 1 and 2014 are divisible by 2 but are not divisible by 5 and 7?

I could help with the following exercise please. How many numbers between 1 and 2014 are divisible by 2 but are not divisible by 5 and 7?
0
votes
1answer
46 views

Big Number representation — Does this exist, and where?

I am at the level of hobbyist, so please tolerate my naivete. I am taking a sophomore-level math course, Discrete Math, and being introduced to some interesting concepts such as Big-O (Time ...
0
votes
1answer
84 views

What does it mean for a “place” to divide?

A proof I'm trying to understand refers to the set of all finite places dividing an algebraic integer x. What does this mean? I can't seem to find a definition in any of the texts I've looked at. ...
0
votes
2answers
116 views

Does sum of all natural numbers contradict another rule?

I must say that I am not a mathematician, just a enthusiast who likes to read all the "weird" results in mathematics. I read that sum of all natural number equals to $-1/12$ and I am also aware that ...
0
votes
2answers
39 views

Distinct pairs with equal sum mod p

Let $p$ be a prime and $\mathbb{F}$ be a field with $p$ elements. Define the sets $$A=\{ (m_1,m_2) : m_1, m_2 \in \mathbb{F}, m_1 \neq m_2 \}$$ and $$T =\{ (a_{1},a_{2}) : a_1, a_2 \in A, a_1 ...
0
votes
2answers
42 views

How do I prove that $x \equiv d \bmod(n\cdot m)$ iff $x \equiv d \bmod n$ and $x \equiv d \bmod m$?

Does this statement hold for any number of factors in the first modulus? One direction is trivial, but I'm stuck trying to prove the other. Any help would be appreciated!
0
votes
1answer
86 views

Diophantine equation has at least $k$ positive integer solutions

$k$ is a positive integer. Could we find all the pairs of positive integers $(a,b)$ such that the Diophantine equation $$x(x+a)=y(y+b)$$ has at least $k$ positive integer solutions $(x,y)$, that is ...
0
votes
2answers
31 views

Does this modular arithmetic equation hold?

Does this modular arithmetic equation hold: $$a_1N_1+a_2N_2+a_3N_3+\cdots+a_mN_m \equiv a_1+a_2+a_3+\cdots+a_m \mod {N_1+N_2+N_3+\cdots +N_m+}$$
0
votes
2answers
55 views

Modular Arithmatic

I have been struggling with modular arithmetic, and I would like to try and finally grasp the concept. In particular, solving problems like $7^{30}$ mod 49. I know I will have to use Fermat's Theorem ...
0
votes
1answer
88 views

How many distinct (non-isomorphic) field extensions of degree n are there?

My study group ha raised an interesting question. Up to isomorphism, how many field extensions of degree n are there? Since a field extension is by definition a vector space, they are all isomorphic ...
0
votes
2answers
68 views

Provide a proof using the rational roots theorem

Prove that if $c$ is a positive rational and $k$ is a positive integer, then $c^{1/k}$ is either an integer or irrational, using the rational roots theorem.
0
votes
1answer
191 views

Square root of 5 in modulo prime field

How can we efficiently find square root of 5 in a mod prime field. By quadratic reciprocity we can argue that 5 is a square in modulo p(prime) is p is square modulo 5. But how exactly can we calculate ...
0
votes
2answers
94 views

How find this value $m^2-mn-n^2$

let $$1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\cdots\dfrac{1}{1}}}}}=\dfrac{m}{n}$$ where $m,n$ are positive integer numbers,and such $gcd(m,n)=1$.,and article $1998$ the fractional ...
0
votes
2answers
53 views

Prove $\frac{3^{2m+1}+1}{4}$ and $3^{2m+1}-1$ are coprime

For a positive integer $m$, let $n=2m+1$. Then how can I prove $\gcd{(\frac{3^n+1}{4}, 3^n-1)}=1$? Thanks in advance.
0
votes
2answers
75 views

how to find the near number to make zero remainder for a quotient?

There is an expression A*x/B, where A and B are given constant (positive integer), x is also positive integer and initially given but I am trying to find the next integer such that A*x is multiple ...
0
votes
1answer
105 views

Show that if $(a, b) = 1$, $a|c$ and $b|c$, then $(a · b)|c$. [duplicate]

"Show that if $\;(a, b) = 1\;$, $\;a|c\;$ and $\;b|c$, then $(a · b)|c$." $$$$Show: We know that $$x\mid w \;\;\text{and}\;\; y\mid w \Longleftrightarrow \frac{x\cdot y}{(x,y)}\mid w$$So if$$a\mid ...
0
votes
2answers
116 views

how many $0$ does $150!$ have when it transform to base $7$?

how many $0$ does $150!$ have when it transform to base $7$? can someone help me please how can I able to solve this problem.thanks for your help.
0
votes
1answer
169 views

Infinitely many primes of the type 5 mod 6.

Problem: Prove that there are infinitely many primes of the type 5 mod 6. My professor did the problem and the proof was horribly long. Can someone show me a shorter version of the proof of this ...
0
votes
3answers
54 views

find $0 < l < 35$ such that $l^5 \equiv 3\pmod {35} $

I have to find some $0 < l < 35$, such that $l^5 \equiv 3\pmod {35} $. I tried to use suggestions from my previous question, So I tried: $l^5 \equiv 3\pmod {35} $ => $35 | l^5 - 3$, I ...
0
votes
1answer
49 views

Show that if $b\mid c$ and $b > \gcd(c, d)$, then $b\nmid d$.

I have no clue about how to go about this question. I feel like I need more info, but I don't know, please help.
0
votes
3answers
106 views

Quadratic residues, mod 5, non-residues mod p

1) If $p\equiv 1\pmod 5$, how can I prove/show that 5 is a quadratic residue mod p? 2) If $p\equiv 2\pmod 5$, how can is prove/show that 5 is a nonresidue(quadratic) mod p?
0
votes
2answers
565 views

Missing Exercises in Elementary Number Theory by Underwood Dudley.

I'm a beginner in math and I just started studying Elementary Number Theory by Dudley. So far I'm impressed, but I've noticed that the book does not include all the solutions to the exercises they ...
0
votes
1answer
51 views

Finding Residue Classes

Can someone explain to me how to find all the elements in $\mathbb{Z}^*_{15}$ Wouldnt the solutions be: $[1], [2], [7], [11], [13]$
0
votes
1answer
53 views

Question regarding Legendre symbol and Quadratic reciprocity.

How would determine the value of the following Legendre symbol is $1$ or $-1$? $$\left(\frac{\frac{p - 1}{2}}{p}\right)$$ So far, I've been able to figure out this much: $$\left(\frac{p - ...
0
votes
2answers
272 views

Mapping from 1D line to 2D plane: an infinite piece of rope covering 2D plane without self-intersection

I believe I'm looking for a function: $f(x) : \mathbb N \mapsto \mathbb N^2$ and it's inverse $f^{-1}(x) : \mathbb N^2 \mapsto \mathbb N$, a known mapping that can take any positive integer and map ...
0
votes
2answers
86 views

Number theory equation(diophantine)!

Find all $x$, $y$ integers s.t. $ 8(x+y)(x^2+y^2)=15(x^2+y^2+xy+1) $ I am new to Diophantine equation, I have tried all kinds of algebraic manipulations but in vain, I have also tried to think of it ...
0
votes
2answers
126 views

How to calculate: $\displaystyle \frac{1}{2}+\frac{3}{4}+\frac{5}{6}+…+\frac{99}{100} $

How can I calculate value of $\displaystyle \frac{1}{2}+\frac{3}{4}+\frac{5}{6}+.....+\frac{97}{98}+\frac{99}{100}$. My try:: We Can write it as $\displaystyle \sum_{r=1}^{100}\frac{2r-1}{2r} = ...