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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1answer
21 views

Why $n^{d(n)/2}$ is not getting satisfied?

Respected all. We know that the product of all positive divisors of $n\in \mathbb N$ is $n^{d(n)/2}$ where $d(n)$ is the number of positive divisors on $n$. What will happen if $d(n)=odd$ say we ...
2
votes
0answers
28 views

Poles and zeroes

If $f$ be the function defined by $$f(x)=2sin\frac{x}{2}\prod_{k=1}^{\infty}\frac{(1-e^{ix}q^k)(1-e^{-ix}q^k)}{(1-q^k)^{2}}$$ where $q = e^{2\pi it}$ $h(x)=\frac{f'(x)}{f(x)}; \quad \quad ...
0
votes
1answer
16 views

Showing Zp is isomorphic to the completion of Z, when Z is considered as a p-adic metric space

Question: Show that Zp is isomorphic to the p-adic completion of Z; that is, the completion of Z when Z is considered a metric space via the p-adic metric. I'm stuck. If we take an element a in Zp, ...
1
vote
2answers
57 views

Diophantine equation resembling FLT

I was wondering if the equation $x^p+y^p=2z^p$ has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular ...
-1
votes
0answers
25 views

Prime Factorization of 6

What would be the prime factorization of 6 in $Q[√−1]$? Can I generalize this to other numbers as well or no? Can someone please help me here?
27
votes
2answers
510 views

Prove that $\frac{a^n-1}{b^n-1}$ and $\frac{a^{n+1}-1}{b^{n+1}-1}$ can't both be prime.

Prove that $$\frac{a^n-1}{b^n-1} \ \text{and} \ \frac{a^{n+1}-1}{b^{n+1}-1}$$ cannot both be prime ($a>b>1,n\ge 2$). Clearly $(a^n-1,a^{n+1}-1)=a-1$ and $(b^n-1,b^{n+1}-1)=b-1$. ...
3
votes
1answer
36 views

What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$?

Theorem #114 in Hardy and Wright says if $p = 4k+3$ then $$ \left[\frac{1}{2}(p-1)\right]! \equiv (-1)^\nu \mod p$$ where $\nu = \# \{ \text{non residues mod } p\text{ less than }p/2\}$. Is ...
0
votes
1answer
30 views

Prove that if p is an odd prime and p does not divide $ac$, then $ax^2+bx+c \equiv 0$ mod $p$ and $cx^2+bx+a \equiv 0$ mod $p$

Prove that if p is an odd prime and p does not divide $ac$, then $ax^2+bx+c \equiv 0$ mod $p$ and $cx^2+bx+a \equiv 0$ mod $p$ have the same number of solutions. Any help would be appreciated.
2
votes
0answers
27 views

Generalization of Dirichlet convolution

The Wikipedia page on the Mobius inversion formula gives the following formula in passing: if $$G(x)=\sum_{k=1}^x \alpha(x)F(x/k)$$ for some arithmetic function $\alpha(n)$ possessing a Dirichlet ...
15
votes
4answers
2k views

$n!+1$ being a perfect square

One observes that $4!+1 =25=5^{2}$, $5!+1=121=11^{2}$ is a perfect square. Similarly for $n=7$ also we see that $n!+1$ is a perfect square. So one can ask the truth of this question: Is $n!+1$ a ...
0
votes
0answers
51 views

When this operation is associative?

I am looking for all possible positive values of $\alpha$ such that the binary operation (on natural numbers) defined by $$m\circ n = mn + \lfloor\alpha n\rfloor\lfloor\alpha m\rfloor$$ is ...
0
votes
1answer
273 views

What's the difference between the euclidean algorithm and the extended euclidean algorithm?

What does the euclidean algorithm compute, and what problems is the extended euclidean algorithm used for? Can someone please show how they each differ on the pair $(210,65)$
1
vote
1answer
35 views

How many infinite subsets of the Naturals have natural density (asymptotic density) zero?

Are there countably or uncountably many? I know that the set of all primes has density zero. Is there an obvious way of using that result to construct an uncountable family of such sets?
2
votes
1answer
22 views

Finite field extensions - $K(\alpha)$

So I am currently studying Algebraic Number theory and a theorem in the Book states the following: Let $L/K$ be a field extension. Then $\alpha \in L$ is algebraic over $K$ if and only if there is ...
1
vote
2answers
28 views

prove that $\forall k\in \mathbb Z$ if $(m,n)=1$(coprimes) then $(k,m,n)=1$

Prove that $\forall k\in \mathbb Z$ if $(m,n)=1$(coprimes) then $(k,m,n)=1$ I did this by contradiction: let $d=(k,m,n)$ so that $d\neq 1$; by definition of greatest common divisor $d|k, d|m, d|n$ ...
-2
votes
0answers
49 views

Cantor's Diagonal: Why not a 1-2 Correspondence between the Naturals and Reals?

Hopefully I'm following Cantor's Diagonal Argument with a minimum of distortion and omission: We start from an enumeration T of all infinite binary sequences. We then construct a list S of elements ...
2
votes
2answers
41 views

Number Theory Primitive Roots Confusion

The following theorem is in my lecture notes: If p is a prime number, then there exist φ(p-1) distinct primitive roots modulo p. I am struggling to make sense of this. φ(m) is the number of ...
2
votes
0answers
47 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1$$ $$x,y = 149,107$$ $$x,y = ...
0
votes
0answers
36 views

Prove $\frac{-5-\sqrt{5}}{2}$ is a quadratic residue when $p\equiv 1\pmod 5$ [on hold]

The number $5$ is a quadratic residue modulo $p$ when $p\equiv\pm1\pmod 5$. How to prove that $\frac{-5-\sqrt{5}}{2}$ is a quadratic residue modulo $p$ when $p\equiv1\pmod5$, and ...
0
votes
0answers
17 views

Monograph about periodic representations of numbers in non-integer bases

I'm looking for a monograph (book, article, lecture notes, whatever) about the representation of numbers (real or complex) in non-integer bases. I am especially interested in results about algebraic ...
3
votes
1answer
33 views

Summing of digits for large numbers

Are there any ways to sum the digits of numbers that are especially big. I have searched for any solutions on stackexchange but could not really find methods that are more informative. For example, ...
1
vote
0answers
42 views

Number of pairs of integers $(a,b)$ with $a^2+b^2=n$ for a constant $n$ [duplicate]

Is there a more general formula for the number of pairs of nonnegative integers $(a,b)$ with $a^2+b^2=n$ for arbitrary $n$?
3
votes
3answers
98 views

Find all prime numbers of the form $n^2 + 4n$

Question: Find all the prime numbers of the form $n^2 + 4n$. List of the primes of this form and prove these are all such primes. My Answer I'm not really good at this but I made an attempt. $$n^2 ...
6
votes
2answers
246 views

RH would follow from $\displaystyle \frac{p_{n+1}}{p_{n+1}-1}<\frac{\log\log N_{n+1}}{\log\log N_n} $ for all $n>1$; what is my mistake?

Let $N_n=\prod_{k=1}^np_k$ be the primorial of order $n$,$\gamma$ be the Euler-Mascheroni constant and $\varphi$ denote the Euler phi function. Nicolas showed that if the Riemann Hypothesis is true, ...
2
votes
5answers
53 views

Possible solutions of a diophantine equation:$p^2+pq+275p+10q=2008$

What are couples of prime integers that verify this diophantine equation:$$p^2+pq+275p+10q=2008?$$ I tried to solve this equation trough the rules of modular-arithmetic. I rewrite the equation as: ...
6
votes
4answers
291 views

How find the postive $m,n$,such $a^n\equiv 1\pmod m$

Find all positive integer pairs $(m,n)$ with $m,n \ge 2$ such that for all $a\in\{1,2,\cdots,n\}$, $$a^n\equiv 1\pmod m$$ If $(a,m)=1$, Euler's Theorem tells us that $$a^{\phi(m)}\equiv 1\pmod ...
1
vote
1answer
43 views

Congruence mod $p$

I need a proof for the following: Suppose that $p$ is an odd prime. If $(a, p) = 1$, then $x^2 = a \pmod p$ either has exactly $2$ solutions or has no solutions within $\textrm{crs}/p$. I can come ...
0
votes
1answer
56 views

Erdős Prime Sieve Conjecture

I think this is/was an Erdős conjecture. I can't find it, or see how to prove it. We know all primes but a finite number can be expressed as $6k\pm1$. If we have a finite set of moduli using ...
3
votes
3answers
60 views

Prime ideals lying above in $\mathbb{Q}(\sqrt{-5})$

I'm really struggling to understand the concept of prime ideals lying above and below a given prime ideal. For example taking the extension $\mathbb{Q}(\sqrt{-5})\big/\mathbb{Q}$, how do we know $(2, ...
1
vote
1answer
33 views

How do I prove that $Z[w]/(a)\cong Z_{N(a)}$?

Define $S=\{\sqrt{D}:D\equiv 2,3 \pmod 4\}$ Define $T=\{\frac{1-\sqrt{D}}{2}:D\equiv 1 \pmod 4 \}$. Let $w\in S\cup T$. Let $N:\mathbb{Z}[w] \rightarrow \mathbb{Z}$ be the norm. Then how do I ...
2
votes
0answers
29 views

For any given value x, are there uncountably many (countably infinite) binary sequences (ones and zeroes) whose limiting relative frequency is x

I have the following question, and given few proofs (provided by friends, professors, and my myself) which seem to work, I suspect the answer is yes: But I am still not completely sure. The question ...
2
votes
2answers
182 views

Arbitrarily large arithmetic progressions only with perfect powers?

How can I show that there are arbitrarily large arithmetic progressions consisting of perfect powers? I think I could use the Chinese Remainder Thm here, but how can I use it? How can I organize the ...
51
votes
5answers
5k views

Is this of any real importance to the mathematical scientific community?

I'm a 31 year old engineer, and I've recently came up with a way to exactly predict the probability of the number of prime numbers between two different integers. For example using my way, the number ...
1
vote
2answers
55 views

Polynomials mod prime $p$

The problem is $5m^2+m+4 \equiv 0\pmod 7$. I am supposed to first convert it to a quadratic whose first coefficient is $1$. But the polynomial cannot be factored, so I am unsure as to how to do ...
4
votes
3answers
80 views

Given an odd $x$ there is an $m,n$ such that $2^n + 1 = 3^m x$?

I'm curious about this question: Is it true that for any odd number $x\in 2\mathbb N + 1$ there exists numbers $m,n\in \mathbb N \cup \{0\}$ such that $$2^n+1 = 3^mx$$ Edit: I'm not trying to make ...
10
votes
1answer
143 views

Integer solutions to $\prod\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}x_i^2$

Given integers $x_1,\dots,x_n>1$. Let's assume WLOG that ${x_1}\leq\ldots\leq{x_n}$. I want to prove that the only integer solutions to any equation of this type are: $x_{1,2,3 ...
0
votes
1answer
37 views

Given integers a; b; c; d;m; n; u; v satisfying ad -bc = 1, u = am + bn, v = cm + dn, prove that (m; n) = (u; v) [closed]

I am unable to solve it can any one help me out please. here it is gcd(m,n) and gcd (u,v)
1
vote
2answers
43 views

Numeric system without “zero”, how to explain importance of zero to average person?

As we all knew that Aryabhata (http://en.wikipedia.org/wiki/Aryabhata#Place_value_system_and_zero) invented zero ($0$) in our number system. I have few questions about it. How did the numeric system ...
2
votes
1answer
371 views

What are the necessary and sufficient conditions for a cubic equation to have integers roots

Let's start with Fermat equation with the lowest power, $x^3 + y^3 = z^3$. Now let's set $y = x + a, z = x + b$ with $b > a$ and $a,b$ integers. then the equation becomes $$x^3 + (3a-3b)x^2 + ...
57
votes
3answers
962 views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
3
votes
1answer
366 views

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem.

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using Gelfond-Schneider's theorem. I'm interested in this problem because I knew that ${\sqrt2}^{\sqrt2}$ is a transcendental number ...
1
vote
1answer
80 views

Diophantine equation $a^3 + b^3 + c^3 = 2$

I have a pretty difficult math question that I have no idea even how to begin. Here it goes: Find the nonzero integers $a$, $b$, $c$ such that $a^3 + b^3 + c^3 = 2$? I would assume that at ...
0
votes
3answers
121 views

Prove that $3 - 2 ^ {1/7}$ is Irrational

How to prove that $3 - 2 ^ {1/7}$ is irrational? If I do $$\frac p q = 3 - 2 ^ {1/7}$$ $$2 ^ {1/7} = 3 - \frac p q $$ Hint needed Should I multiply by $7$ times??
2
votes
0answers
63 views

Can Waring problem be solved with triangle inequality?

When we calculate the difference $\frac{3^k-1}{2^k-1}-\left(\frac{3}{2}\right)^k,$ we get $\frac{3^k-2^k}{4^k-2^k}.$ Then solving: $1-\frac{3^k-2^k}{4^k-2^k}>\frac{3^k-2^k}{4^k-2^k},$ we get ...
5
votes
1answer
27 views

Show that the ring of integers $A$ of the cubic field $\mathbb Q[x]$ with $x^3=2$ is principal.

Show that the ring of integers $A$ of the cubic field $K=\mathbb Q[x]$ with $x^3=2$ is principal. The hint given in the book is to majorize the discriminant of $A$ by $D(1,x,x^2)$ and then use the ...
1
vote
3answers
86 views

Can transcendental to the power transcendental be rational?

Can a transcendental number to the power of a transcendental number be a rational number?
-3
votes
0answers
41 views

Prove every even integer greater than $2$ can be expressed as the sum of two primes. [closed]

I need help finding a proof for this statement: Every even integer greater than $2$ can be expressed as the sum of two primes. My attempt so far: I have proved that the statement is ...
2
votes
1answer
52 views

Number Theory : Is a complete residue system modulo $n$ a group?

I was working my way through some basic number theory problems, when in the chapter on "Introduction to Group Theory," I came across the following: Show that for every positive integer $n$, the ...
0
votes
3answers
61 views

Find all solutions mod $19$ to $4x^2+6x+1 \equiv 0$ mod $19$

Find all solutions mod $19$ to $4x^2+6x+1 \equiv 0$ mod $19$ I am not really sure how to start this problem.
4
votes
3answers
92 views

Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ a number field?

Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ of $5$-adic numbers a number field, if yes what is the degree ? To be honest I don't understand the question, what does it mean ...