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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18 views

A problem about number of congruence classes in a set

Let us consider a prime, $\ p $,for which we choose $\ s $ different integers,$\ a_1 $ ,$\ a_2 $...$\ a_s $ where $\ s$ < $\ p $ , such that they all are coprime to $\ p $ and belong to different ...
0
votes
0answers
21 views

Using hensel's lemma

I dont know why if f'(1)=0 mod3 , then f(1)=36 mod81 means there is no solution. we dont need to test each step? In mod3 's sold considering sol of mod9 And with sol of mod9. We decide sol of mod27 ...
0
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0answers
19 views

The multi-binomial theorem, prove multiples of 11

Over $\mathbb{Z_{11}}$ $ $, $ \\\\$ $f(x)= x^{11} - x $ has solutions $0, 1, 2, \cdots , 10.$ (by Wilson) $$$$ So, we can rewrite $f(x)=x(x-1)(x-2)(x-3)\cdots(x-10)$ That is $$x^{11} - x = ...
1
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4answers
75 views

Find the sum to $n$ terms of the series $10+84+734+…$

Find the sum to n terms of the series $10+84+734+....$ $\frac{9(9^n+1)}{10} + 1$ $\frac{9(9^n-1)}{8} + 1 $ $\frac{9(9^n-1)}{8} + n $ $ \frac{9(9^n-1)}{8} + n^2$ My attempt: I'm getting option ...
0
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0answers
40 views

What is $\stackrel{d}{=}$?

What does $\stackrel{d}{=}$ mean? I see it in this sort of context: $$ \operatorname{Var}_x\omega \stackrel{d}{=} (1/x)\sum_{n\le x}(\omega(n)-\bar\omega_x)^2 \sim \log\log x $$ which in this case ...
1
vote
2answers
24 views

Cubic modular equation

If i found the sol of (a), i can find sol of (b) and (c) by using hensel's lemma i want to know the way of solving (a) without testing all the case (ex testing 1,2,3,4,...10) I think that there ...
1
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1answer
37 views

The exponent of $11$ in the prime factorization of $ 300!$ is___.

The exponent of $11$ in the prime factorization of $ 300!$ is $27$ $28$ $29$ $30$ My attempt: According to Exponent of $p$ in the prime factorization of $n!$ ...
2
votes
3answers
79 views

Is $53\cdot 83\cdot109+40\cdot66\cdot96$ prime or composite?

Let $$A=53\cdot 83\cdot109+40\cdot66\cdot96$$ Is this number prime or composite? I'm sure it's a composite number. But I do not know how to prove it.
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3answers
54 views

Is it true: $3k^2+1$ is a perfect square if and only if $k=1$ or $4$

I'm checking the following conjecture: $3k^2+1$ is a perfect square if and only if $k=1$ or $4$. If it is not true counter example would be appreciated. Thanks in advance.
0
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1answer
33 views

What is this series/function called? (and is it converging)?

An interesting question popped into my mind a few hours ago, specifically.. what is the relation some number $n$ and the numbers behind it. Clearly $n$ is 1 bigger than $n-1$ and 2 bigger than $n-2$.. ...
3
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3answers
33 views

Proof that $\sum_{d|m} |\mu(d)|=2^n$, where $n$ is the number of distinct prime divisors of $m$?

Given an integer $m$ such that $n$ is denoting the distinct prime divisors of $m$, is there a proof that the sum over the divisors of m of the absolute value of the Möbius function $\mu(d)$ is equal ...
2
votes
1answer
32 views

Basis of a Cyclotomic Field

I've started learning algebraic number theory when I found something that confused me; for a prime $p$, where $\zeta=e^{(2\pi i/p)}$, a primitive $p$-th root of unity. Then the extension ...
0
votes
2answers
29 views

Partition of integers with distinct primes

Is there a number P such that every sufficiently large integer can be written as the sum of at most P different primes?
2
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0answers
34 views

Diophantine System Solution

Could you please help with finding of general solution of diophantine system for rational a, b, c, d $(a^2+b^2)(c^2+d^2)=A^2$ $(a^2-b^2)(c^2-d^2)=B^2$ for some rational A and B. This is related ...
0
votes
3answers
110 views

A sum of irrational numbers is an algebraic integer

I am asked to show that $$\sqrt{2} + \sqrt[3]{5} - \sqrt{17} \Big(\frac{7 - \sqrt{13}}{2} \Big)$$ is an algebraic integer. $\textbf{Definition:}$ An algebraic integer is the root of a monic ...
5
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1answer
72 views

If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$?

If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$? I know we want to use Hensel's Lemma somehow to assess this question, but I'm ...
4
votes
1answer
64 views

Exists sequence converging to $0$ in $\mathbb{R}$, $1$ in $\mathbb{Q}_2$?

Does there exist a sequence of elements $x_1, x_2, x_3, \ldots$ of elements of $\mathbb{Q}$ that converges to $0$ in $\mathbb{R}$ and converges to $1$ in $\mathbb{Q}_2$?
4
votes
3answers
82 views

Motivation for rings of fractions?

I'm learning about rings of fractions and localization. I like the material a lot and feel engaged with it, but I do lack a broader perspective on things. For example, I'm aware of things such as ...
0
votes
1answer
26 views

If G is the quotient group $\mathbb{Q}/\mathbb{Z}$ show that mG = G for any $m \epsilon \mathbb{N}^+$

Def: $mG$ = {$ma$ | $a \epsilon$ G} To show $mG = G$ I can show $mG \subseteq G$ and $G \subseteq mG$. To show $mG \subseteq G$: Take $x \epsilon mG$. Then $x$ has the form $m$($\mathbb{Z} + ...
3
votes
1answer
69 views

Sum of products of $(1 - 1/p)$

Let $\pi(n)$ denote the number of primes not greater than $n$, and $p_k$ the $k$th prime, so that $p_{\pi(n)}$ denotes the largest prime not greater than $n$. I'm interested in the value of the ...
2
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2answers
37 views

If the sequence $ x_{n} $ converges to L, then $\lim_{k\to \infty}x_{k+1} = L $

Can someone read this proof and let me know if it is correct? If the sequence $ x_{n} $ converges to $L$, then $$\lim_{k\to \infty}x_{k+1} = L $$ Proof. Let $ \epsilon > 0$, and suppose ...
3
votes
1answer
62 views

Comparison of $ ( 1^a + 2^a+ … n^a)^n$ and $n^n(n!)^a $

For a given real number $a>0$ , define $ d_n =( 1^a + 2^a+ ... n^a)^n $ and $ b_n = n^n(n!)^a $ for $ n = 1,2,\ldots$ Then a) $ d_n< b_n $ for $ n> 1$, b) There exists an integer ...
4
votes
3answers
59 views

Quick, self-contained way to see why $\left({{-1}\over p}\right) = 1$?

Let $p$ be a prime number congruent to $1$ modulo $4$. What is a quick and self-contained way to see why$$\left({{-1}\over p}\right) = 1?$$
0
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0answers
12 views

Convergence of Sums With Logarithmic Numerator

Let $S \subseteq \mathbb{N}$ have the property: $\displaystyle \sum_{n=1}^\infty \frac{\ln (s_n)}{s_n}$ converges. I'm just wondering if there are any known theorems out there which allow us to derive ...
-2
votes
1answer
73 views

Find all primes n such that $n^3+1$ is a perfect square [closed]

Find all primes $n$ such that $n^3+1$ is a perfect square
1
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2answers
24 views

If G is the quotient group $\mathbb{Q}/\mathbb{Z}$ show that G has exponent $0$.

Let G = $\mathbb{Q}/\mathbb{Z}$, and show that G has exponent $0$. I don't see how $0$ would be the exponent, because if I am understanding the definitions correctly this would say that each coset in ...
1
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0answers
16 views

Regulator of an algebraic number field and regulator of units.

Let $K$ be an algebraic number field. I would like to know why in totally real fields regulator is at most equal to regulator of a set of $n-1$ independent units. What happens in the case of $K$ not ...
2
votes
1answer
45 views

Given $p$ prime for some $p$ deduce $2p+1$ prime

Given $p=33179$ and $2^{2p+1}\equiv 2\; \pmod{2p+1}$, deduce $2p+1$ is prime. All I can think of is using Fermat's little theorem: $2^{2p}\equiv 1\pmod{2p+1}$ which just tells me it may be prime.
0
votes
1answer
21 views

Theorem precedding Schnirelmann's Theorem

An Introduction to Sieve Methods and their applications has the following Theorem 6.3.4 $$|\lbrace p\leq x:\;|p+\alpha| \text{ is prime} \rbrace|<\frac{cx}{(\log x)^2} ...
3
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0answers
33 views

On the limit $\lim_{n \to +\infty} n \{ n \xi \}$

Assume that $\xi \in \mathbb{R} \setminus \{Q\}$ is a given irrational number. I am trying to draw some conclusion about the limit $$ \lim_{n \to +\infty} n \{ n \xi \} $$ where $\{\cdot\}$ denotes ...
3
votes
1answer
104 views

Find the sum of $1^{n}-2^{n}+3^{n}-4^{n}+\cdots+m^{n}$

After seing this question I started wondering about a generalization of a similar sum. The sum is $$ S(m,n)=\sum_{r=1}^{m}(-1)^{r-1}\;r^{n} $$ I gave this to WA to crunch and it gave $$ S(m,n)= ...
0
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0answers
39 views

How to generate the sequence of prime building blocks of the colossally abundant numbers: $2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 23, 2,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers: $2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, 7, 29, 3, 31, 2, 37, ...
-8
votes
3answers
96 views

The dilemma of Pi [closed]

Is Pi rational or irrational ? Pi can be represented as 22/7 which is a rational number. Whereas 3.14 is a non terminating and non recurring number which is a irrational number
-1
votes
1answer
33 views

How to rigourously prove that any integer divisible by $3$ can be written as a sum of four cubes? [closed]

How to rigourously prove that any integer divisible by $3$ can be written as a sum of four not necessarily posiitive cubes? I have been trying it for long
0
votes
1answer
8 views

$g$ is a primitive root of $p^s$, determine the form of all solutions to $x^{p-1} \equiv \pmod p^s$

$g$ is a primitive root of $p^s$, then all the solutions of the congruence $x^{p-1}\equiv 1 \pmod p^{s}$ are given by $1, g^{p^{s}},\ldots, g^{p^{s}(s-2)}$ Clearly that the given set of solutions fit ...
6
votes
1answer
145 views

Is there a (not so) generalized version of Hilbert's Theorem 90?

I'm sorry if my following question doesn't make any sense. We know that if $L/k$ is a finite Galois extension then $H^{1}(\mathrm{Gal}(L/k),L^{*})=0$ (Hilbert's theorem 90). However I would like to ...
1
vote
1answer
50 views

For which odd primes $p ≠ 5$ is 10 a qudratic residue modulo $p$?

For which odd primes $p ≠ 5$ is 10 a quadratic residue modulo $p$? Saw a similar example using 5 and 15 and did my best to learn from those but still having a hard time grasping how to complete this ...
1
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4answers
76 views

Determine whether $x^2 - 14x + 30 \equiv 0$ mod 1615 is solvable. If so, find its solutions…

Determine whether $x^2 - 14x + 30 \equiv 0\pmod{1615}$ is solvable. If so, find its solutions. I assume the best way to solve this is via Chinese Remainder Theorem, but first i would have to break ...
1
vote
1answer
40 views

Suppose that gcd($a,p$) = gcd($b,p$) = 1, and neither of the congruences $x^2 … [duplicate]

Suppose that gcd($a,p$) = gcd($b,p$) = 1, and neither of the congruences $x^2 \equiv a$ mod $p$ or $x^2 \equiv b$ mod $p$ has a solution. Show that $x^2 \equiv ab$ mod $p$ does have a solution. ...
0
votes
2answers
16 views

computing value of binary string

How do we compute the value of binary string S[0..9] modulo 3 efficiently if we know the corresponding value for S[0..4] and S[5..9] ?Let us say we have a binary string s 1010001010 of length 10 then ...
1
vote
1answer
17 views

Proving a statement with inequality and min operator

How can I prove/disprove the following statement. $$\min\{m_1+m_2,p_1+p_2\} \geq \min\{m_1,p_1\} + \min\{m_2,p_2\}$$ Where $m_i$ and $p_i$ for $i=1,2$ are natural numbers. My intuition says that ...
-1
votes
0answers
23 views

Finding order of solution to a system of congruences

I'm working on this problem: I'm not really sure where to start. I'm pretty sure I should use the Chinese remainder theorem. Can you help me out?
0
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2answers
42 views

solve $x^2 -4x +13 \equiv 0 \pmod{81}$?

How do I solve $x^2 -4x +13 \equiv o \pmod{81}$ ? I know that this is the same as $x^2 -4x +13 \equiv x^2 + 2x + 1 \equiv (x +1)^2\equiv 0\pmod{3^4}$ but why is $x \equiv -1\pmod{3}$ the only ...
1
vote
2answers
26 views

Help solving a question using Quadratic Reciprocity?

How do i solve this equation using Quadratic reciprocity? How many solutions does the quadratic equation $\bar{x}^{2} = \bar{2}$ have in $\mathbb{Z}_{47}$? I have no idea how to go about this i ...
0
votes
1answer
36 views

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 -y^2 = 2000^2$

I found this answer here on AoPS. I agree with the answer till it multiplies $49$ by $2$. I think it should be multiplied by $4$ since there are $4$ possible cases: 1) $x+y, x-y$ is positive. 2) ...
1
vote
5answers
62 views

Does excluding or including zero from the definitions of “positive” and “negative” make any consequential difference in mathematics?

I was absolutely certain that zero was both positive and negative. And zero was neither strictly positive nor strictly negative. But today I made a few Google searches, and they all say the same ...
1
vote
1answer
48 views

How many numbers $m$ satisfy $1 ≤ m ≤ n$ and $\gcd (m, n) = 1$?

Let $n = p^2 q$ where $p$ and $q$ are distinct prime numbers. How many numbers $m$ satisfy $1 \leq m \leq n$ and $\gcd (m, n) = 1$? Note that $\gcd (m, n)$ is the greatest common divisor of $m$ and ...
6
votes
1answer
109 views

Solve $x^2 = 2^n + 3^n + 6^n$ over positive integers.

Solve $x^2 = 2^n + 3^n + 6^n$ over positive integers. I have found the solution $(x, n) = (7, 2)$. I have tried all $n$'s till $6$ and no other seem to be there. Taking $\pmod{10}$, I have been ...
-4
votes
1answer
69 views

First scientific work. [closed]

One year ago I decided to myself to write my own scientific work in number theory, graph theory or combinatorics. I tried to find the teacher and theme during this year, but unfortunately I didn't ...
2
votes
2answers
67 views

RSA - Proof for dummies

I'm understanding the basic idea behind why RSA is secure, but I'm having a hard time understanding its proof with only basic knowledge of numbers theory. so I'm hoping that somebody can help me ...