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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Decomposition of cyclotomic extensions with totally real components

Let $\mathbb{Q}(\xi_n)$ be a cyclotomic extension. For which values of $n$ is it the case that this extension of $\mathbb{Q}$ decomposes as (direct sum of) extensions of $\mathbb{Q}$ such that at ...
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1answer
34 views

how do put 64 squares on a cube?

I need 64 squares covering a cube. The amount of squares on each on each face must be equal to each other. The amount of squares on each must be a natural number. The root of the amount of squares on ...
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0answers
33 views

what is zebloski's number? [on hold]

I have been searching for quite some time now.this question was an interview question.i am curious.it was asked by a maths professor in an examination.i have googled it and also searched through some ...
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0answers
57 views

An equation which generates all primes within a specific range

Does there exist an equation which generates all primes within a specific range like 10 to 100 ? If I discover one such kind of equation, will it be a good discovery ?
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1answer
31 views

Every element of the form $x^n - \beta$ is a norm?

Let $F$ be a local $p$-adic field containing the $n$th roots of unity. The notes I'm reading claim that every element of the form $x^n - \beta$, for $x, \beta \in F$, is a norm from ...
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0answers
8 views

Existence question about Hamming weights of addition of numbers modulo $2^n-1$

Let $w_1, w_2$ be given, $0 \leq w_1, w_2 \leq n-1$. Given an integer $a$, $1 \leq a \leq 2^n-2$, can we find $b$ with $W_H(b) = w_1$ such that $W_H(a + b \mod{2^n-1}) = w_2$? Here $m \mod 2^n-1$ ...
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0answers
28 views

The ring of modular forms for $\Gamma_0(11)$

Let $\mathcal M(11) = \oplus \mathcal M_k(11)$ be a graded algebra of modular forms for congruence group $\Gamma_0(11)$. I want to find generators and relations between them. I proved that $\dim ...
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1answer
28 views

How do you convert different bases?

I know how to convert any number into base 10 by using the below method. Write (6712)base 8 in base 10. Ans: $6 \times 8^3 + 7 \times 8^2 + 1 \times 8^1 + 2 \times 8^0 = 3530_{10} $ However, I am ...
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1answer
18 views

Is this Bertrand's postulate-related statement valid?

Bertrand's postulate says: For every $n>1$ there is always at least one prime $p$ such that $n<p<2n$. Is the following statement: For every $n>3$ there is always at least one ...
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1answer
25 views

All possible variants of representation natural number N as product of natural numbers

Task : describe a predicate (on Prolog) that count all possible variants of representation of natural number N as product of natural numbers. For example, 6 = 6*1 = 2*3, so answer is 2. The program ...
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18 views

Why is $K^{\ast n}$ contained in the norm group?

http://www.bprim.org/cyclotomicfieldbook/rlmain.pdf In section 5, $K$ is a local $p$-adic field containing the $n$th roots of unity, and $L = K(\sqrt[n]{x} : x \in K^{\ast})$. Kummer theory tells us ...
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1answer
33 views

Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$

I'm currently reading Andreescu and Andrica's Number Theory: Structures, examples and problems. Problem 1.1.7 states : Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$. The ...
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3answers
58 views

How do I find the factorial of a decimal [duplicate]

How do I find the following: $$(0.5)!(-0.5)!$$ Can someone help me step by step here?
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0answers
13 views

Special $\omega(n)$-sequence

Let $k$ be a natural number, $\omega(n)$ the number of distinct prime factors of $n$. The object is to find a number $n$ with $\omega(n+j)=j+1$ for each $j$ with $0\le j\le k-1$. In other words, a ...
2
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1answer
29 views

Lower bounds on the index of $\mathbf Z[X]/(P)$ in the ring of integers of a number field

Let $P$ in $\mathbf Z[X]$ be an irreducible polynomial. Let $\mathcal O$ be the ring of integers of the number field $K:=\mathbf Q[X]/(P)$ and $i$ be the index of $\mathbf Z[X]/(P)$ in $\mathcal O$. ...
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37 views

If $m^n -m=(m-n)!$, where m>n>1 and $ m=n^2$, then the value of $m^2 +n^2 =?$

If $m^n -m=(m-n)!$, where m>n>1 and $ m=n^2$, then $m^2 +n^2 =?$ how can I find this one ?? I found ridiculous after some stapes..
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1answer
18 views

Quadratic Diophantine equations on the ring of polynomials

The set of solutions of quadratic equation $a^2+b^2=c^2$ on $\mathbb{Z}$ can be described by Pythagorean triples up to multiplication. Can I use similar results on the ring of integer coefficient ...
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3answers
67 views

Integers of the form $x^2+2y^2$.

I'm stuck in the following problem: prove an integer $n$ is of the form $x^2+2y^2$ if and only if every prime divisor $p$ of $n$ that is congruent to $5$ or $7\bmod8$ appears with an even exponent. I ...
5
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1answer
39 views
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6answers
34 views

$\gcd(a,n)\neq 1 \implies $ there is $b$ such that $ab\equiv 0 \pmod{n}$

I have that $\gcd(a,n)\neq 1$ ($a$ and $n$ are not coprime). Then, somehow, I need to prove that exists an $b$ such that $$ab\equiv 0 \pmod{n}$$ What I did: $$ab\equiv 0 \pmod{n}$$ is the same ...
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2answers
35 views

What delimits the mathematical framework within which information compression limits (from entropy) are valid.

Lets suppose for absurd that I eliminate one number from the naturals. If I were supersticious I would eliminate number 13. Now imagine that to keep normal mathematics possible within such system ...
5
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1answer
57 views

Extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, splitting.

In the extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, must principal prime ideals of $\mathbb{Z}[\sqrt{-5}]$ necessarily split into 2? Must nonprincipal prime ideals not split? ...
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29 views

Convergence of series involving Euler's totient function.

I have to show that if $\phi$ is Euler's totient function, then the series $\sum\limits_{n=2}^{\infty} \frac{1}{\phi (n) \log n}$ diverges and $\sum\limits_{n=2}^{\infty} \frac{1}{\phi(n) \log^2 n}$ ...
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0answers
30 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
5
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1answer
109 views

Prove by combinatorial method that $ \frac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $ is an integer [duplicate]

Prove that $$ \dfrac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $$ is a positive integer, where $(m,n) \in \mathbb{Z^{+}}$ I have already solved it using Legendre's Formula ...
2
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1answer
40 views

for any positive integer $a,b,n$,and $(a,b)=1$,Is $\frac{1}{a+b}+\frac{1}{a+2b}+\cdots+\frac{1}{a+nb}$ non integer,and How to prove that?

It's easy to prove that both $\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ and $\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2n+1}$ are nonintegers by multiply $2^k$and $3^k$, and how about the ...
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1answer
55 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
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17 views

On (known) applications of fixed point theorems to some conjectures in elementary number theory

Let $\sigma$ be the classical sum-of-divisors function. Call an integer $n$ almost perfect if $\sigma(n)=2n-1$. The only known examples are $n=2^k$ for $k \geq 0$. Let $I(n)=\sigma(n)/n$ be the ...
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1answer
60 views

Why are there so many conjectures in number theory and comparatively less in others?

My question is that : Why are there so many conjectures in elementary number theory and comparatively less in others? This is particularly weird because every topic in maths should have its equal ...
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1answer
58 views

Why is $\frac{4k -1}{2} \equiv 1 \pmod 4$?

Why is $\frac{4k -1}{2} \equiv 1 \pmod 4$? I need some help with understanding this... The original problem was: find the set of Numbers in which $ f(z)=\frac{1}{1- sin(z)} $ is not defined. This ...
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0answers
17 views

On Diophantine approximation and irrationality proofs

This question is an offshoot from this previous MSE post. I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / ...
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2answers
51 views

How to show that $(2, \sqrt{82})$ in $\mathbb{Z}[\sqrt{82}]$ is not pricipal?

I tried the obvious things, like using the norm and trying to show that there were no integer solutions to $a^2 - 82b^2 = 2$, but didn't get anywhere. (A friend asked me this.)
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2answers
67 views

Finding de Dr. Bronowski's number

This is a puzzle, created by Dr. J. Bronowski. All credit for this problem goes to him. Find the least positive integer such that moving the leading digit to the end produces a new integer that is ...
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1answer
27 views

unramified quadratic extensions of 2-adic numbers

i already know how to get the 7 quadratic extensions of $\mathbb{Q}_2$ from hensel's lemma. they are $\mathbb{Q}_2(\sqrt{d})$ for d = -10, -5, -2, -1, 2, 5, 10. question: which of these are ...
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1answer
69 views

Finding all primes $p$ such that $3p+20$ and $p+20 $ are primes

I need to find all primes $p$ such that $3p+20$ and $p+20 $ are primes. The first primes which satisfy the condition are: $3, 11, 17, 23$. I've tried to find the dependence, and found that $p \equiv ...
5
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1answer
78 views

theorem relating mersenne numbers?

For $(x2^9)^2=2^q-1+y^2q^2$,where $q$ is prime, is it possible to show that there exists only an unique solution for the pair $\{x,y\}$?
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1answer
39 views

Number of primes between $2k$ and $(\sqrt{k}-1)^2$.

I would like to prove the following. Let $\pi$ denote the prime counting function. Then for $k\geq 81$ we have $$ \pi(2k)-\pi\left(\left\lfloor(\sqrt{k}-1)^2\right\rfloor\right)\geq 6. $$ What I ...
5
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2answers
84 views

$3^x + 4^y = 5^z$ [duplicate]

This is an advanced high-school problem. Find all natural $x,y$, and $z$ such that $3^x + 4^y = 5^z$. The only obvious solution I can see is $x=y=z=2$. Are there any other solutions?
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0answers
24 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general?

(Note: This question has been cross-posted from MO.) Let $\sigma$ be the classical sum-of-divisors function. A number is said to be perfect if $\sigma(N)=2N$. If $q^k n^2$ is an odd perfect number ...
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1answer
50 views

A number-theoretic random walk on the integers

Suppose a random walker starts at $S_0 = 2$, and walks according to the following transition probabilities: If the walker is on the $n$th prime number $p_n$, she moves to either $p_n + 1$ or ...
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0answers
19 views

Is there a classification of perfect squares in the rings of cyclotomic integers?

We know that perfect squares in $\mathbb {Z} $ has the form $n^2, n\in \mathbb {Z} $. More generally, can I use a nice condition to find a solution of Diophantine equations of the form $x^2=c $, $c ...
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0answers
44 views

Proving the existence of a sequence such that

I am trying to prove that there exists a sequence, for example: $$ f(n) = n! $$ (or we can select any sequence we need to prove the existence of just one), with the following property: edit: for ...
2
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2answers
49 views

Proof that $a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$

I need to prove that: $$a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$$ What I thought was: $$a\mid x \implies x = aq_1\\b\mid x\implies x = bq_1$$ Also, since $\gcd(a,b) = 1$, we have that ...
2
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2answers
65 views

$\lfloor 2x \rfloor \lfloor 3x \rfloor$ [on hold]

From $1$ to $10000$ including both, how many of those integers can be written as: $$\lfloor 2x \rfloor \lfloor 3x \rfloor$$ Where $x$ is a real number?
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totient function and inclusion-exclusion principle

How can one prove the property established by Gauss $$ \sum_{d\mid n} \varphi(d)=n$$ using the inclusion–exclusion principle? I was thinking to use that with the same method one can prove that ...
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3answers
74 views

Show $ x^2 = 1 + y^2 + z^2$ has infinitely many solutions [on hold]

Show $ x^2 = 1 + y^2 + z^2$ has infinitely many solutions Can anyone give me the specific steps for this problem?
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2answers
80 views

Finding two solutions to $x^2 - 6y^2 = 1$ using continued fractions [on hold]

Can anyone show me how to find the solutions to $x^2-6y^2=1$ by using continued fractions? I know how to find the fractions for $\sqrt6$ but do not know how to proceed. THANK YOU!!!
5
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3answers
929 views

Where is the mistake in this proof about rationals having eventually periodic decimal expansions?

I know that a rational number has an eventually periodic decimal expansion, and not necessarily just periodic. So what is wrong with this 'proof' that any rational number has a periodic decimal ...
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0answers
92 views

ABC triples when C is a power. [on hold]

For any integer $x$ we can apparently always find a sufficiently large $y$ such that there is an $ABC$ triple $A + B = C$ (from the $ABC$ conjecture $rad(ABC) < C$) where $C = x^y$. The minimum ...
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2answers
51 views

A question about Quadratic residue

I need help with this question : Prove that for each prime number p there exist $a,b \in Z$ such that $-1\equiv a^{2}+b^{2}\pmod p $ When $p\equiv1\pmod4$ it is easy because -1 is a quadratic ...