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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10 views

arbitrarily long sequences without perfect powers

The fact that there are arbitrarily long sequences of consecutive numbers without prime number is well known, the proof is easy and goes like this: let $n\ge 2$, then the number $n!+k$ is greater ...
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1answer
15 views

Product of Quadratic Residues

If a is a quadratic residue, and ab is a quadratic residue, how can I show that b is also a quadratic residue? Would appreciate a hint. So far I thought about the problem a little and I have: $a^2$ ...
9
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0answers
44 views

Numbers that are clearly NOT a Square

Although I have never studied math very seriously, I have heard of Brocard's Problem, which asks for integer solutions for the following Diophantine Equation:$$n!+1=m^2$$ The only solutions are ...
15
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5answers
545 views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
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1answer
40 views

number pair's in the self-root function $f(x) = x^{1/x}$

in the self-root function $f(x) = x^{1/x}$ the output is in pairs of numbers i.e. $f(2) = f(4)$ , the inputs are 2 apart producing the same output , the square root 2 is equal to the 4th root of 4 ...
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1answer
38 views

What are the invariants of a number field? [on hold]

How is defined an invariant of a field? Given a certain field extension $L/K$, is it related with the Galois group ${\rm Gal}(L/K)$? In the case of number fields, which are the invariants associated ...
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2answers
11 views

$a =\prod_{i = 1}^{r} p_{i}^{a_i}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$…

$a =\prod_{i = 1}^{r} p_{i}^{a_i}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$,Prove that $a$ is the square of an integer if and only if $a_i$ is even for each $i$. -The ...
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1answer
14 views

$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$…

$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$, deduce a formula for the sum of the squares of the positive divisors of $a$. -The section we ...
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0answers
49 views

Any good math books? [on hold]

I was wondering if there are any books about a bunch of random math theories, areas and topics. For example topics like group theory, randomness. klein bottles and other interesting things
3
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2answers
37 views

Let $C$ be the set of all complex numbers of the form $a+ b \sqrt {5}i$, where $a$ and $b$ are integers…

Let $C$ be the set of all complex numbers of the form $a+ b \sqrt {5}i$, where $a$ and $b$ are integers. Prove that $7$, $1 + 2\sqrt {5}i$, and $1 - 2\sqrt {5} i$ are all prime in $C$. -I am really ...
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3answers
46 views

Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$

Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$ I was a bit lost with this proof until I found a similar looking proof-based question from a previous ...
1
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1answer
70 views

A problem of decimals..

The exact problem: For any natural number n>1, write the infinite decimal expansion of $\frac{1}{n}$ (for example, we write 1/2 = $0.4\overline9$ as it's infinite decimal expansion, not 0.5). ...
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3answers
226 views

What is wrong with this infinite sum [on hold]

We know that: https://www.youtube.com/watch?v=w-I6XTVZXww $$S=1+2+3+4+\cdots = -\frac{1}{12}$$ So multiplying each terms in the left hand side by $2$ gives: $$2S =2+4+6+8+\cdots = -\frac{1}{6}$$ This ...
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0answers
12 views

Is there K and an infinite amount of different primes $a_i,b_i$ so that min|$a_i^y-b_i^x$| <K on natural x,y for all i?

First of all I know that it was proved recently that prime gaps are less than like 7 million for an infinite amount of primes, but I'm not smart enough to follow the proof. I am looking for a ...
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3answers
42 views

What is the significance of using prime numbers in proving: $x$ is a multiply of $y$?

I came to a problem where it asks me to prove, for example, $n^4-n^2$ is a multiple of $12$. Now, factorize the multiple: $n\times n\times (n-1)\times (n+1)$. Here we have $3$ consecutive integers. ...
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0answers
19 views

$k$ be a positive integer ; $\{a_n\}$ be a sequence of positive integers such that $\gcd(m,n)=1 \implies a_m^k+a_n^k|m^k+n^k$ ; then $a_n=n$?

Let $k$ be a positive integer ; $\{a_n\}$ be a sequence of positive integers such that g.c.d.$(m,n)=1 \implies a_m^k+a_n^k|m^k+n^k$ ; then is it true that $a_n=n , \forall n \in \mathbb N$ ?
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0answers
42 views

Is it true that for every positive integer $m$ , there exist a positive integer $n$ such that $\phi(n)=m! $ ?

Is it true that for every positive integer $m$ , there exist a positive integer $n$ such that $\phi(n)=m! $ ?
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4answers
39 views

No solutions to diophantine equation

I am trying to deduce that $x^2-5y^2=0$ having shown that $x^2 \equiv 5y^2 (mod 7)$ has no integer solutions (not 0). How do I go about this?
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1answer
125 views

Is there any palindromic power of $2$?

My question is in the title: Is it possible to find $n≥4$ such that $2^n$ is a palindromic number (in base $10$)? A palindromic number is a number which is the same, independently from which ...
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2answers
48 views

Prove Expression cannot be factored

I am currently working on primes which can be expressed in form of a polynomial. For eg, Find all primes which can be expressed in form $n^4-52n^2+595$ It is very essential to tell whether a ...
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0answers
30 views

Find the minimum number of tickets to guarantee the win of a n-bit binary lottery?

Here's the problem. I just don't know how to approach it. If the 'one error tolerance' were removed, then this would be a simple binomial distribution problem. But now I can't figure it out. In ...
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0answers
28 views

Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes?

This question follows from the information provided below. Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial ...
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2answers
81 views

Investigating Nicolas' criterion for the Riemann Hypothesis. [on hold]

Throughout this note, $N_k$ denotes the $k$-th primorial number (the product of the first $k$ primes), $\varphi(n)$ the Euler totient function, and $\gamma$ is the Euler-Mascheroni constant. By the ...
2
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1answer
57 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 ...
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1answer
25 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 ...
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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 ...
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3answers
31 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 ...
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0answers
55 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) = ...
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1answer
63 views

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

Where can I find the best proof that $e$ is transcendental?
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1answer
40 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.
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1answer
43 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
40 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 ...
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0answers
30 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 ...
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0answers
22 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
81 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
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4answers
41 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.
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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?
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1answer
44 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
27 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
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1answer
71 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 $x,y$, $|3^x-17^y|>2$.
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1answer
33 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 $z+_n a$ ...
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0answers
45 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
28 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 ...
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1answer
70 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
38 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
24 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
36 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 ...