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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25 views

There are infinitely many odd numbers that are not expressible as the sum of a power of 2 and a prime

I have to prove this but so far I have struggled to find an example of an odd number that never gives a prime when powers of two are subtracted from it. Can someone give me some hints and/or examples ...
4
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2answers
381 views

Is PA the first axiomatization of arithmetic to be discovered? [closed]

Is Peano Arithmetic the first axiomatization of arithmetic to be discovered?
3
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0answers
43 views

Sum of integer squares with zero sum

This is something that I perhaps should know but don't. What is known about sums $\sum_{i=1}^k a_i^2$ subject to $\sum_{i=1}^k a_i=0$ where $a_i$ are integer? Specifically, which even integers can be ...
0
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0answers
23 views

Factorising into Gaussian primes $4+3i$

I want to find the Gaussian prime factors of $(4+3i)$ $$(4+3i)(4-3i) = 25=5^2$$ $$5: (2-i)(2+i)$$ so $$(4+3i)(4-3i) = 25=5^2 = (2-i)(2+i)(2-i)(2+i)$$ That was my answer but the solution says: ...
2
votes
2answers
60 views

If primitive root modulo $mn$, then primitive root modulo $m$ and $n$

Let $a$ be a primitive root modulo $mn$. Show that $a$ is also primitive root modulo $m$ and $n$. Showing $(a,mn)=1\Longrightarrow (a,m)=(a,n)=1$ is not a problem. The problem is showing $a^{\varphi ...
0
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0answers
18 views

Definition of $S$-ideles

This is a basic notational question. Let $K$ be a number field and $M_K$ the set of all places of $K$ with $S\subset M_K$ a finite subset. Write $\mathfrak J_K$ for the idele group of $K$ and ...
0
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1answer
30 views

If an integer $k$ is a divisor of an integer $n$, then $\frac{n}{k}$ is also a divisor of $n$?

A lot of these number theory ideas are popping up in my study of cyclic groups. In particular, in a note I came across. It is mentioned that: If an integer $k$ is a divisor of an integer $n$, ...
1
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1answer
56 views

Continued Fractions : Under which branch of mathematics do they come?

I wanted to know in which branch of Mathematics do Continued Fraction come? By branch I mean for example Geometry or Differential Equation are a branch of maths so is there any particular branch of ...
1
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3answers
54 views

How do I show that $\gcd(n,\frac{n}{k})=\frac{n}{k}$?

$\gcd(n,\frac{n}{k})=\frac{n}{k}$ Let $n, k$ be positive integers. This should be a trivial question but not having taken any classes in number theory I would like to be convinced with a simple ...
1
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1answer
24 views

Gcd of a Numbers

What is the Sum of GCD of this:where G is gcd of two numbers G(1,x)+G(2,x)+G(3,x)+..G(x,x)
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0answers
21 views

Prove that 4 is not a primitive root modulo n, for $ n \ge 2$ [duplicate]

I want to prove that 4 is not a primitive root modulo n, for $ n \ge 2$ I know how to prove it for any prime n, but I don't know how to prove it for numbers bigger than 2 which are not primes. ...
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25 views

Unique base $10$ representations of real numbers

Prove that the base $10$ representation of any real integer value is unique. More generally prove that the base $n$ representation of any real integer value is unique. This seems obvious to me, ...
3
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2answers
31 views

Followup to question on $5$-adics, if $k \in \mathbb{Q}_5^\times$, is there $x_1, x_2, x_3 \in \mathbb{Q}_5$ where $\sum_{i = 1}^3 x_i^2 = k$?

This is a followup to my question here. My question is as follows. If $k \in \mathbb{Q}_5^\times$, then are there $x_1$, $x_2$, $x_3 \in \mathbb{Q}_5$ where$$x_1^2 + x_2^2 + 3x_3^2 = k?$$My idea is ...
3
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2answers
31 views

$p$-adics, elements of $\mathbb{Q}_5^\times/(\mathbb{Q}_5^\times)^2$?

Here is a question surrounding the $p$-adics. I am curious as to what the description of the quotient group $\mathbb{Q}_5^\times/(\mathbb{Q}_5^\times)^2$ is, i.e. what are its elements? Here is an ...
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2answers
30 views

find the maximum value of some Summation

Let A be a groups of some numbers: $$A = \big\{x_{1},x_{2},...,x_{n}\big\}$$ such that for any 1<= i <= n: $$0<= x_{i} <=1$$ and $$\sum_{i=0}^n x_{i} = 1$$ There is a formula for the ...
3
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1answer
74 views

Fix $k∈\Bbb{N}$. An integer $n$ is said to be $k$-th power free if there exists no prime $p$ such that $p^k | n$…

Fix $k∈\Bbb{N}$. An integer $n$ is said to be $k$-th power free if there exists no prime $p$ such that $p^k | n$. Prove that for any $m∈\Bbb{N} $ greater that 1, there exists $m$ consecutive integers ...
1
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1answer
32 views

Show that 3 is a primitive root modulo 14. Then, write the other primitive roots modulo 14 in terms of powers of 3. How many are there? [closed]

Show that $3$ is a primitive root mod $14$. Then, write the other primitive roots mod $14$ in terms of powers of $3$. How many are there? A bit lost with this question. Poked around online and ...
1
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1answer
22 views

Let $p ≥ 3$ be a prime number, let $r ∈ N$, and let $x$ be a primitive root modulo $p^{r}$. Show that $x$ is a primitive root modulo $p$.

I want to prove this: Let $p ≥ 3$ be a prime number, let $r ∈ N$, and let $x$ be a primitive root modulo $p^{r}$. Show that $x$ is a primitive root modulo $p$. Well I don't have any ideas, I know ...
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1answer
28 views

If p is an odd prime, compute $S = \sum^{p-1}_{k=1} (\frac{k}{p}) $ and $P = \prod^{p-1}_{k=1} (\frac{k}{p})$… [closed]

If $p$ is an odd prime, compute $S = \sum^{p-1}_{k=1} (\frac{k}{p}) $ and $P = \prod^{p-1}_{k=1} (\frac{k}{p})$, the sum and product of all nonzero Legendre symbols modulo p. Honestly have no idea ...
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2answers
35 views

Prove that $40\mid 3^{4N}-1$

Prove that $40 \mid 3^{4N}-1$ for all integers $N$. How would I prove this by using modular arithmetic? I can't remember how to prove this. Any help would be appreciated.
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0answers
23 views

Finding a primitive root modulo $p$ [duplicate]

Let's attempt to find a primitive root modulo, say, $p=127$. Since $p$ is prime a primitive root exists (more specifically there are $\varphi (\varphi (127))=36$ primitive roots modulo $127$). ...
0
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0answers
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
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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
20 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 = ...
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4answers
78 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 ...
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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
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2answers
25 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
38 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
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3answers
80 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
55 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
35 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
33 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
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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
37 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
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3answers
112 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
votes
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
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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
votes
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
61 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
13 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 ...
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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
vote
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
18 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} ...