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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0
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1answer
18 views

How to find open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$?

For purposes of illustrating Local Class Field Theory, let us play with the $3$-adic numbers. I'd like to find some open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$. I know about the ...
1
vote
1answer
27 views

Can every rational number be represented as the product of four rational numbers with an additional property?

Suppose that $q$ is a rational number. Are there such $a,b,c,d\in \mathbb{Q}$ that: 1) $a\cdot b\cdot c\cdot d = q$; 2) $a+b+c+d = 0$?
4
votes
1answer
130 views

Permutation Partition Counting

Consider the number $n!$ for some integer $n$ In how many ways can $n!$ be expressed as $$a_1!a_2!\cdots a_n!$$ for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as ...
2
votes
2answers
474 views

What is the average prime numbers we've found till now?

When you count from 0 to 100 you have 25% prime numbers. Till now the largest prime consists of $2^{74,207,281}-1$ numbers. But is known what the average is till now? With average I just mean the ...
2
votes
1answer
34 views

There are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is a Euclidean field

My book's exercise is about proving that there are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is a Euclidean field (with respect to the norm). ...
-2
votes
0answers
17 views

if p and q are distinct primes and n=p q then there is a primitive root mod n

if p and q are distinct primes and n=p q then there is a primitive root mod n could you help me with counterexample I prove it and I find this statement is false , I try many example and all of them ...
14
votes
1answer
243 views

Find a closed form formula for $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes.

I meant by "closed form formula" a formulate that doesn't have summation or has very few terms. Maybe there's a better term for this meaning. I found this function that has very interesting property ...
1
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2answers
37 views

Prove that $p|1+2(p-3)!$

Prove that $p|1+2(p-3)!$ I know the wilson's theorem and started with that but I reach a stage where I am not able to solve. $1+(p-1)!= M(p)$ $=1+(p-1)(p-2)(p-3)!= M(p)$ $=1+ (p^{2}-3p+2)(p-3)! ...
0
votes
0answers
12 views

Is there application of sieve method $\{x: x \neq \pm 1 \text{( mod $p$)}, x \in \mathbb{Z}, p \le p_i\}$ for twin primes?

I'm studying the twin prime numbers. Instead of sieving prime numbers, I found this method to sieve $\{x: x \neq \pm 1 \text{( mod $p$)}, x \in \mathbb{Z}, p \le p_i\}$, so that $(x-1,x+1)$ will be ...
1
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2answers
24 views

How to calculate $x$ in $19^{93}\equiv x\pmod {162}$?

I have to calculate $19^{93}\equiv x\pmod {162}$. All I can do is this,by using Euler's Theorem:- $19^{\phi(162)}\equiv1\pmod{162}$ So,$19^{54}\equiv1\pmod{162}$ Now,I have no idea how to reach ...
7
votes
2answers
439 views

Fermat's Last Theorem's Proof

Where can I find a copy of the proof that proves Fermat's Last Theorem? Also, what different mathematical topics would I need to know to fully understand all of the syntax and reasoning in the proof? ...
1
vote
1answer
13 views

If p>2 is prime and r is primitive root mod p then r^((p-1)/2) == -1 (mod p)

If p>2 is prime and r is primitive root mod p then r^((p-1)/2) == -1 (mod p) Could you help me this statement is true or false ? I do it by fermat little theorem and I find it equal + - 1(mod p) I ...
-1
votes
1answer
22 views

If $a$ is odd then$(a^2)^n \equiv1$ (mod $2^{n+1}$) for all $n \geq 1$

If $a$ is odd then $(a^2)^n \equiv 1 (mod 2^{n+1})$ for all $n \geq 1$ I know it is false statement when I prove it by induction but could you help me give me counterexample show it is false I try ...
3
votes
2answers
45 views

Given $x^3$ mod $55$, find its inverse

So i am wondering how i can figure out what the functional inverse of $x^3$ mod $55$ is. I can only assume it is $x^{1/3}$ mod $55$ but i am not sure if that is the form i should keep it in
5
votes
0answers
32 views

Is there a clever way to find a smaller number that produces the Euclidean algorithm of given length?

Is there a simple way to tell if for a given $n$ there is $m$ such that the Euclidean algorithm on $n,m$ runs for a given number of steps, and/or a way to find $m$ efficiently (other than testing all ...
1
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0answers
25 views
3
votes
1answer
71 views
+50

Pell's equation and representation elements of $\mathbb Z_p$.

We defined the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $c\not\equiv 0\pmod{p}$. Is it true that $f$ is onto?
33
votes
0answers
1k views

Do $p,q$ exist such $|p-q|+|a_{p}-a_{q}|=2014$

Let $\{a_1,a_2,\ldots,a_{2016}\}=\{1,2,3,\ldots,2016\}=A$ be such $$\dfrac{a_i-a_j}{i-j}\neq 1,\forall i,j\in A\text{ with } i\neq j.$$ Show that there exists $p,q\in A$ such that ...
3
votes
3answers
113 views

Find all natural numbers $n$ such that $21n^2-20$ is a perfect square.

Find all natural numbers $n$ such that $21n^2-20$ is a perfect square. I have got the following solutions via programming: $n=1,2,3,9,14,43,67$ but how can I find these manually? How can I ...
2
votes
0answers
26 views

Let $N=3^{1000}\times 2^{200009} +1$. Show that $5^{\frac{N-1}{2}}\equiv -1 \pmod{N}$.

This is showing that 5 is a quadratic non-residue mod N but I don't get why this says it is prime. The question also asks that you say that if p was prime divisor of N what the power of 2 dividing ...
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0answers
27 views

Number of solutions to a modular equation of a specific form

I struggle with this Exercise, or at least the part where one should prove how many solutions there are: Let $p$ be an odd prime, and let $e\in\mathbb{Z}$ with $e\gt 1$. Let $a$ be an integer of ...
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votes
0answers
21 views

Norm of ideal belongs to the ideal [duplicate]

Suppose that $D$ is any number ring (i.e. $D=\mathbb Q(\alpha), \alpha \in \mathbb C$). Let $I$ be any ideal of $D$. Show that $N(I)=|D/I|$ belongs to $I$. How to start? is there a specific fact will ...
0
votes
3answers
59 views

How many roots does this polynomial have in $\mathbb{Z}/91\mathbb{Z}$?

$f(x)=x^8-1$ I know how I would do this problem if the mod wasn't so high. Not sure how to approach this question.
6
votes
0answers
49 views

Which prime factors of $8^{8^8}+1$ are known?

We have the partial factorization $$8^{8^8}+1=(2^{2^{24}}+1)\cdot (2^{2^{25}}-2^{2^{24}}+1)$$ The first factor is $F_{24}$. It is composite, but no prime factor is known. A prime factor of the ...
0
votes
0answers
8 views

Factoring the conductor of a Dirichlet character and factoring generalized Gauss sums

Let $m=m_1m_2\cdots m_r$ with $m_i$ positive integers with $\gcd(m_i,m_j)=1$ for $i\neq j$. Given a Dirichlet character $\chi$ modulo $m$ we can define the characters $\chi_i$ (modulo $m_i$) by ...
0
votes
0answers
4 views

Particle moving and dividing on real line, reference to Sierpinsky triangle

Problem: At time $t=0$, a particle is at $x=0$ on the real line. At $t=1$, the particle divides into $2$ and moves one unit to the left and the other moves one unit to right. At $t=2$, each of these ...
7
votes
0answers
35 views

Sum of three consecutive cubes equals a perfect square

I have found this problem in an old German textbook: Find all sets of three consecutive integers such that the sum of their cubes is a perfect square. We can write $$S = (x-1)^3 + x^3 + (x+1)^3 = ...
1
vote
2answers
29 views

$2^{49}$ ways to choose a set of integers $\leq 50$ with odd sum

Problem: Show that the number of ways one can choose a set of distinct positive integers, each smaller than or equal to $50$, such that their sum is odd, is $2^{49}$. My attempt: Suppose set ...
0
votes
1answer
6 views

Distances between identical strings in a long Vigenere

My queston is "Distances between identical strings in a long Vigenere ciphertext are 18, 30, 12, 12, 18. What is the likely key length"? I'm looking in the book and it has a similar problem that ...
0
votes
1answer
40 views

Using an exponential cipher system, encipher the word HALT. where $p = 29, k = 11$, and $m = 1$.

Using an exponential cipher system, encipher the word HALT. where $p = 29, k = 11$, and $m = 1$. The progress I have made so far: H A L T $07, 00,11,19$ Since, $m =1$, we break this up into $2*m$ ...
8
votes
3answers
142 views

Finding all solutions for $3^c=2^a+2^b+1$

Given the equation: $3^c=2^a+2^b+1$ Find all solutions for $a,b,c$ - given that they are positive integers and $b>a$. Any ideas?
0
votes
0answers
9 views

Recursive definition of Minkowski ?(x) function

There is a fact that $?(\frac{a+b}{c+d}) = (?(\frac {a}{c}) + ?(\frac{b}{d})) / 2$ if a/c and b/d are adjanced elements of Farey sequence. How to prove it? I don't have any ideas at all.
29
votes
0answers
276 views

Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, K, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n$$ ...
0
votes
2answers
23 views

Calculate a quadratic irrational from its periodic continued fraction

I have a periodic continued fraction [2; 1, 3] and I want to convert it into a quadratic irrational. Any helps?
1
vote
1answer
54 views

Is it known if $\frac{\zeta(3)}{\pi^3}\in\mathbb{Q}$? [duplicate]

Is it known if $\frac{\zeta(3)}{\pi^3}\in\mathbb{Q}$? It is obvious that $\frac{\zeta(2n)}{\pi^{2n}}\in\mathbb{Q}$, but since there is no closed form for the odd values, are we left to be unable ...
3
votes
1answer
457 views

What are the necessary and sufficient conditions for a cubic equation to have integers roots

Let's start with Fermat equation with the lowest power, $x^3 + y^3 = z^3$. Now let's set $y = x + a, z = x + b$ with $b > a$ and $a,b$ integers. then the equation becomes $$x^3 + (3a-3b)x^2 + ...
9
votes
3answers
226 views

Find all integral solutions for the Diophantine Equations $x^4 - x^2y^2 + y^4 = z^2$ and $x^4 + x^2y^2 + y^4 = z^2$.

Find all integral solutions for the Diophantine Equations $$x^4 - x^2y^2 + y^4 = z^2$$ and $$x^4 + x^2y^2 + y^4 = z^2$$ I basically think that to solve these equations we need to use the fact ...
0
votes
0answers
15 views

Intuition in understanding Minkowski question mark ?(x) function

There are 3 definitions of Minkowski function: 1) If $[a_0; a_1, ...]$ is a continuous function representation of n 2) Consider the different ways of interpreting an infinite string of bits ...
0
votes
0answers
23 views

For a given non-constant polynomial $f(x)$ with integer coefficients, how many solutions are there to $f(x)\equiv 0 \mod(n)$ where $n$ is composite?

For a given non-constant polynomial $f(x)$ with integer coefficients, how many solutions are there to $f(x)\equiv 0 \mod(n)$ where $n$ is composite? Is there a general way to determine the number of ...
0
votes
0answers
19 views

Number of solutions to $f(x)\equiv 0 \mod(11\cdot 19^{2})$

I have been asked to explain why the number of solutions of the polynomial congruence $f(x)\equiv 0 \mod (11\cdot 19^{2})$ cannot be 121, where $f(x)=x^{10}+10x^{8}-17x+12$. Any ideas?
1
vote
1answer
18 views

Is a divides infinitely many repunits?

Let (a,10)=1 Let n=9k{phi(a)} using eulerphi function k is positive integer. When (a,9)= 1 , 3 it is okay Because 81 and a divides 10^n-1 by Binomial theorem and CRT So a divides (10^n-1)/9 ...
5
votes
1answer
35 views

Solve $2^{99}$ mod $101$

My number theory is a bit rusty, so i am trying to recall how to work this problem out. I know that the euler theorem would state that $2^{\phi(101)} \equiv 1$ mod $101$ But in this case, ...
0
votes
1answer
30 views

Have any property of square to make below problem reasonable? [on hold]

$26^2+97^2=62^2+79^2$. How is it possible? Any magic theory involved in there?
2
votes
1answer
30 views

If $p>5$ is prime, $2p+1$ is a prime, $\frac{4p+1}{3}$ is prime, $8p+1$ is prime, Then $p \equiv 29 (mod \; 30)$

Assume that $p>5$ is prime, $2p+1$ is a prime, $\frac{4p+1}{3}$ is prime, $8p+1$ is prime. Then I want to prove that $p \equiv 29 (mod \; 30).$ First of all I have to show that $4p+1$ is a ...
1
vote
3answers
56 views

Order of $5$ in $\Bbb{Z}_{2^k}$

Is it true that the order of $5$ in $\Bbb Z_{2^k}$ is $2^{k-2}$? I was unable solve the congruence $5^n\equiv 1\pmod {2^{k}}$ nor see why $5^{2^{k-2}}\equiv 1\pmod {2^{k}}$. I'm not sure if this is ...
3
votes
1answer
55 views

Simplify $\gcd{(b-a,c-a)}$

Is there any way to simplify $$\gcd{(b-a,c-a)}$$ where all $a,b,c$ are integers? Is there any other way to write/simplify this?
0
votes
1answer
21 views

Fermat's Little theorem (Num Theory) [on hold]

How can I compute $31^{1209}\equiv \mod (101)$ using Fermat's Little Theorem?
0
votes
1answer
38 views

The theory of riemann zeta function titchmarsh page 15 question in the proof of the functional equation

I am currently reading Titchmarsh's book about the Riemann Zeta function and came across a problem in a proof of the functional equation that I cannot solve. To be precise, I am referring to this ...
0
votes
2answers
26 views

Basic Number Theory (Divisibility)

Not sure where to start. Thank you in advance! Find all positive integers $n$ such that $12$ divides $n$ and $n$ divides $816$.
0
votes
1answer
19 views

Multidimensional Cantor diagonal argument for ordering infinite sets [duplicate]

Cantor diagonal argument is a powerful proof technique. It has been used for a lot of proofs. For instance, it has been used to prove that $|\mathbb{N}| < |\mathbb{R}|$. What can we say about the ...