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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1
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0answers
16 views

Norm restricted to $\mathbb Q$

Consider an algebraic extension $K$ of $\mathbb Q$ and suppose that on $K$ we have a non trivial norm $||\cdot||$. Is it possible to have that the restriction $||\cdot||_{\mathbb Q}$ is the trivial ...
0
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0answers
10 views

Incipit of chapter VI of Neukirch's ANT book.

The title of the chapter VI of the neukirch's ANT is "Global class field theory", and the first few lines are the following: the author doesn't explain what is $K$ here, but from the previous ...
1
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2answers
23 views

Find all $7$-digit numbers which use only the digits $5$ and $7$ and are divisible by $35$

Find all $7$-digit numbers which use only the digits $5$ and $7$ and are divisible by $35$. Attempt It is easy to see that all numbers of this form must be of the form _ _ _ _ _ _ 5. Working ...
0
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3answers
28 views

Positive integer not a power of 2

It's given that if a positive integer $n$ is Not a power of two, then $n$ must have an odd prime factor, meaning $$n = pr, p>2, 1\leq r< n $$ Is it really this trivial? There's a proof that ...
-7
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0answers
43 views

Has Brochard's Problem been solved? I'm sure it has… [on hold]

I had a conversation with him, and Wan Chan solved it. Proof: Let $n! +1 = m^2$; $n! = m^2 -1$; $n! = (m +1)(m -1)$. Let $m +1 = k$, and $n! = k*(k -2)$. Thus, for $n = 4$, $4! = 1*2*3*4 = 6*(6-2) = ...
0
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1answer
54 views

I need a best proof that e is a transcendental? [on hold]

Where can I find the best proof that $e$ is transcendental?
0
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1answer
37 views

If $p = a^2 + b^2$, prove that $(ab^{-1})^2 \equiv -1 \pmod{p}$

Let $p \equiv 1 \pmod{4}$ be a prime, where $p = a^2 + b^2$. Show that $(ab^{-1})^2 \equiv -1 \pmod{p}$ I'm having trouble with this question. Any help is appreciated.
0
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0answers
24 views

Does the product of two numbers with a primitive representation have a primitive representation?

I know the theorem that $n = x^2 + y^2, \, \textrm{gcd}(x, y) = 1 \iff p | n \implies p \equiv 1 \bmod 4$. We call an expression of $n$ in this form primitive. I'm trying to prove the statement. I've ...
4
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0answers
26 views

AMM 2488: Primitive Root Relatively Prime to p-1

(from American Mathematical Monthly, problem 2488. I hope this hasn't been posted before but I'm new and maybe not very good at using the search function effectively..) Let $p>3$ be a prime. Show ...
1
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0answers
29 views

How to show that a set of elements is a basis for the ring of integers of a number field?

Let $K$ be a number field of degree $n$ (that is $[K:\mathbb{Q}]=n$) with ring of integers $\mathcal{O}_K$. I know that there exists algorithms to find $\mathcal{O}_K$ and hence determine a ...
1
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0answers
17 views

On an inequality involving primorial numbers.

Let $N_k$ denote the $k-th$ primorial number. That is, the product of the first $k$ primes and $\phi(n)$ be the Euler totient function. How can one show that there exists a constant $\theta>1$ ...
13
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1answer
74 views

Pythagorean triplets of the form $a^2+(a+1)^2=c^2$ and the space between them

I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions: $0^2+1^2=1^2$ $3^2+4^2=5^2$ $20^2+21^2=29^2$ $119^2+120^2=169^2$ ...
2
votes
4answers
36 views

Showing that Harmonic numbers are $\Theta(\log n)$, intuitively

I wish to verify that Harmonic numbers $H_n = \sum_{k=1}^{n} \frac{1}{k}$ are $\Theta(\log n)$. One idea I have is to approximate the sum with an integral: $$\int_{1}^{n} \frac{1}{k} ~dk = \log(n) - ...
0
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2answers
27 views

Congruence problem $12x\equiv3\pmod{45}$ [on hold]

$$12x\equiv3\pmod{45}$$ Find all possible solutions to above congruence and show procedure in detail.
1
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1answer
23 views

Is solvability of diophantine equations over a p-adic field decidable?

As far as I understand, the decidability of solvability of diophantine equations over the rationals is an open problem. What about the decidability of solvability over a given p-adic field?
1
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1answer
43 views

Show that $st$, $(s^2-t^2)/2$ and $(s^2+t^2)/2$ are relatively prime.

Let $s$ and $t$ be odd integers. Show that $st$, $(s^2-t^2)/2$ and $(s^2+t^2)/2$ are relatively prime. I've seen this question on here, but unfortunately some of the cases were not covered, and I ...
0
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0answers
23 views

speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$

I would like to find the speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$ for any $z$. For example, if $|z| < 1$, doing taylor expansion we know that it is roughly $e^{zn}.$ But I need it ...
1
vote
1answer
50 views

Given M, can we find $2$ primes $a,b$ so that for all naturals $x,y$, $|a^x-b^y|>M$?

For example, if $M = 2$, one can show that $3,17$ satisfy the above: For any naturals $x,y$, $|3^x-17^y|>2$.
0
votes
1answer
30 views

Is/when is this property about the totative subset of $(\mathbb{Z}_n , +_n)$ true?

Is it possible to show (or when is it true); that for the group $\mathbb{Z}^+_n :=(\mathbb{Z}_n , +_n) $, there exists an $a \in \mathbb{Z}^+_n$ for each $z \in \mathbb{Z}^+_n$, where both $z+_n a$ ...
0
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0answers
42 views

Can this number be rational?

Let $K = e^{-\gamma}\log\log n$, where $\gamma$ is the Euler-Mascheroni constant and $n\geq 2$ is a positive integer. Can $K$ be rational for any integer $n\geq 2$ ? I seem not to find any argument ...
0
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0answers
26 views

How common are diophantine equations for which the local global principle is invalid?

The local global principle says that in some families of diophantine equations the solvability over the rationals is equivalent to solvability over the reals and in p-adic fields $Q_p$ for each prime ...
2
votes
1answer
31 views

Show that $g^p (1 + p)$ is a primitive root modulo $p^e$

Given that g is a primitive root modulo $p$, show that $g^p (1 + p)$ is a primitive root modulo $p^e$. I'm not really sure where to go with this. the $ gcd(p^e, g^p (1 + p))$ is easy enough to show ...
1
vote
1answer
67 views

Show that the only nonzero ideals of R are the principal ideals $\langle p^e \rangle$

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ \textrm{ord}_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. Show that the only nonzero ideals of $R$ are the ...
0
votes
1answer
31 views

Solving a quartic congruence modulo 175

The congruence I'm trying to solve is $x^4 \equiv 71 \pmod{175}$. I really have no idea how to approach this as 175 isn't a power of a prime.
2
votes
2answers
34 views

On the relationship between $\phi(n)$ and $\sigma( n)$

I recently learnt that $\frac{\sigma(n)}{n} \leq \frac{n}{\phi(n)}$, were $\sigma(n)$ denotes the divisor function, $\phi(n)$ the Euler totient function and $n\geq 2$ is an integer. My questions is ...
2
votes
0answers
23 views

Choosing three integers to satisfy an equation under a specific condition

Find three integers $(a,b,c)$ such that: $x*a + y*b + z*c = a + b$ only when $x = 1, y = 1, z = 0$ where $x, y$ and $z$ can be chosen as any non-negative integers. For example, choosing $a = 1$; $b = ...
0
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0answers
30 views

Regarding the iteration of sum of prime factors

Let $sopf(n)$ be the sum of prime factors of $n$, with repetition for prime factors. I have observed an interesting phenomena when $n$ is a prime number $p$. So for any prime number $p_1$, if ...
1
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0answers
51 views

Test if a number is in ${\mathbb R}$ [on hold]

Given a number $x$ $\in$ ${\mathbb R}$ is there a way to know if $x$ $\in$ ${\mathbb N}$ without comparing $x$ with a number in any known list of numbers? (ex. {0,1,2,3,...}) To be more specific: ...
8
votes
4answers
372 views

Decimals of the square root of $n$.

Let $a_1, \ldots, a_k$ be any sequence of digits (i.e., each $a_i$ is between 0 and 9). Prove that there exists an integer $n$ such that $\sqrt{n}$ has its first $k$ decimals after the decimal point ...
4
votes
0answers
29 views

Proving the congruence of a Fibonacci Number [on hold]

Let $F_n$ denote the $n^{th}$ fibonacci number where $F_0 = 0, F_1 = 1$. Prove that for all primes $p > 5$, $$F_p \equiv 5^{\frac{p-1}{2}} \mod (p)$$
0
votes
2answers
35 views

To find composite integers satisfying the given property.

Find all positive composite integers $n$ greater than $1$ such that for any relatively prime divisors $a$ and $b$ of $n$ with $a > 1$ and $b > 1$, the number $ab-a-b+1$ is also a divisor of $n$. ...
5
votes
1answer
67 views

Help to solve $\displaystyle \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} dz $

I need help in evaluating the following contour integral: $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} ds $$ It looks like a complicated ...
2
votes
0answers
28 views

Is there a short proof of the existence of $a$ so that $a$ is a primitive root for infinitely many primes $p$?

After looking for a general answer I found Artins conjecture, and I was happy to see so much is known. However I don't know nearly enough to follow the proof, yet it bothers me I can't prove the ...
0
votes
1answer
80 views

Ideals of $ord$

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ ord_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. (a) Show that the only nonzero ideals of $R$ are the ...
1
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0answers
30 views

Asymptotic sums and big-O notation

Suppose I have to compute the following asymptotic sum ($x\rightarrow\infty$): $$ S(x):=\sum_{n\leq f(x)} O(g(x,n))\;, $$ where the function $g(x,n)$ is non-decreasing in $n$, so that in our case ...
0
votes
0answers
51 views

Hypothetical proof of Goldbach's conjecture? [on hold]

Goldbach's conjecture: Every even number greater than 4 is the sum of two prime numbers. An equivalent statement is; For all $n\geq 2$ there exists a number $e(n)$ such that $n-e(n)$ and ...
3
votes
1answer
25 views

How to see that the prime gaps functions isn't monotonic?

Let $g(n)$ be the distance between the $n$th prime and the next. By elementary means we can see that $g(n)$ is not eventually constant and that $g(n)$ is not strictly monotonic. Further we know that ...
3
votes
0answers
64 views

Primes and irreducibles of $\{a+b\sqrt{-2}\ |\ a,b \in \mathbb{Z}\}$

Let $R = \{a+b\sqrt{-2}\ |\ a,b \in \mathbb{Z}\}$ Rational primes $p \geq 3$ of the form $p = a^2 + 2b^2$ factorize in $R$ as a >product of two irreducibles which are not associate. Such ...
0
votes
1answer
32 views

Are there any known asymptotics for $\sum_{p\leq x} p$? [duplicate]

As a prospective undergraduate who has really benefited from his time on MSE thus far, i recently learnt that there exists asymptotic approximations for $\sum_{p\leq x} 1, \sum_{p\leq x} p, ...
0
votes
0answers
26 views

Is there a proof that $(n-2x)! \times n^{2x-1} > n!$ (where $x$ is a function related to the prime counting function

Is it possible to prove the following? Let $\pi$ be the prime counting function and $A(n)=\pi(2n)-\pi(n)$ $(n-2A(n))! \times n^{2A(n)-1} > n!$
5
votes
1answer
52 views

Express a real number as a product

Hi guys if I have a number $x \in [1,2)$ is it possible to express such number as: $$x = \prod_{j=0}^{+\infty} (1 + \alpha_j 2^{-j})$$ where each $\alpha_j \in \left\{-1,0,1\right\}$? If yes, how ...
4
votes
2answers
69 views

Prime counting function; when is it true that $\pi(n) > \pi(2n) -\pi(n)$?

Let $\pi$ be the prime counting function. Under what conditions is it proven true that $\pi(n) > \pi(2n) -\pi(n)$, if at all?
0
votes
0answers
56 views

For what values of $a, n$ the number $2^a\cdot 3^n+1$ is prime?

Respected All. Please forgive me as I am unable to understand if it is an off topic. I am stuck in a situation. Change the title if you find inappropriate What I am willing to know is that if there ...
0
votes
1answer
16 views

Writing a product of transpositions as a 3-cycle.

For the case that we have two transpositions equal to each other, say (a b) (a b) then how can I write the product as a product of 3 cycles?
3
votes
1answer
23 views

Quotient & Remainder of $\frac{a}{bc}$

$a, b$ and $c$ are integers: $a,b,c \in \mathbb{Z}$ $b$ and $c$ are greater than zero: $b>0 \wedge c>0$ $q$ and $r$ are the quotient and remainder of $\frac{a}{b}$: $a=qb+r$ $t$ and $s$ are ...
0
votes
0answers
46 views

PID and irreducibles

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ ord_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. (c) Show that the only nonzero ideals of $R$ are ...
2
votes
1answer
40 views

Subrings of the product of rings of algebraic integers

Let $K,L$ be a number fields and $\mathcal{O}_K,\mathcal{O}_L$ their rings of algebraic integers, and $n_K=[K:\mathbb Q]$, $n_L=[L:\mathbb Q]$. Suppose that $R$ is a subring of ...
12
votes
1answer
69 views
+300

Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree

An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree ...
0
votes
1answer
45 views

Question about 'Knuth's up-arrow notation'

Am I right? $$x\uparrow x=x^x$$ $$x\uparrow\uparrow x=x^{x^x}$$ $$x\uparrow\uparrow\uparrow x=x^{x^{x^x}}$$ $$x\uparrow\uparrow\uparrow\uparrow x=x^{x^{x^{x^x}}}$$ ...
7
votes
1answer
66 views

Given $k$, are there infinitely many $n$ so that $w(n) = w(n+k)$?

$w(n)$ denotes the number of distinct prime factors of $n$. I am wondering if any such result is known.