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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Show $x^2+2x+1\equiv 27 \;\text{mod}\; 61$ is solvable and find the number of solutions.

I'll show how far I have got: $$x^2+2x+1\equiv 27 \;\text{mod}\; 61$$ $$(x+1)^2\equiv 27\; \text{mod} \; 61$$ So we need to find the Legendre symbol value for $$\begin{pmatrix} 27\\ 61 \end{pmatrix}$$ ...
2
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2answers
33 views

Find the prime factorization of $X^3-5X^2+6X+7$ in $(\Bbb{Z}/11\Bbb{Z})[x]$.

Find the prime factorization of $X^3-5X^2+6X+7$ in $(\Bbb{Z}/11\Bbb{Z})[x]$. I try doing this with analogy to the integers but I think I am not sure what is prime and what is isn't. I could really ...
3
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1answer
88 views

Numbers that can be expressed as the sum of two cubes in exactly two different ways

It seems known that there are infinitely many numbers that can be expressed as a sum of two positive cubes in at least two different ways (per the answer to this post: Number Theory Taxicab Number). ...
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2answers
22 views

When is $2$ a quadratic residue mod $p$?

For which prime numbers $p$, is $2$ a quadratic residue modulo $p$. I know that $2$ is a quadratic reside iff $$2^{\frac{p-1}{2}} =1 \; \bmod \;(p) $$ so $$2^{p-1} =1 \; \mod \; (p). $$ But I ...
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2answers
24 views

Prove that if $n\in\Bbb{N}$ is odd, and $\phi(n)$ is a power of 2,then $n$ is a product of distinct primes.

Prove that if $n\in\Bbb{N}$ is odd, and $\phi(n)$ is a power of 2,then $n$ is a product of distinct primes. I didn't really think I understood this question. What is the condition exactly? That in ...
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2answers
16 views

A question in Number Theory about Euler Theorem/Fermat little theorem

I tried to solve this question but without a success. for every prime number $$p\ge7 $$ and every $$n \in \mathbb N$$ : $$10^{n(p-1)}\equiv 1 (\text{mod }9p) $$ I tried to use Euler theorem. but It ...
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4answers
181 views

Will the Fermat's last theorem still hold if algebraic and transcendental numbers are introduced?

This might seem as a wild thought and is also a wild thought that will the Fermat's last theorem still hold if algebraic and transcendental numbers are introduced? What I mean to say is that will ...
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4answers
27 views

Prove that if $p\ge 7$ then $\exists n\in\Bbb{Z}$ such that $10^{n(p-1)}\equiv1 \mod 9p$.

Prove that if $p\ge 7$ then $\exists n\in\Bbb{Z}$ such that $10^{n(p-1)}\equiv1 \mod 9p$. Edit: $p$ is prime, of course. I tried using theorems regarding Euler, but I can't seem to arrive at something ...
0
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1answer
52 views

If you know $N=a^2+b^2$ how to compute $a$ and $b$ for large $N$?

Having tested that $N$ is such that every exponent of a prime in the prime factorisation of $N$ congruent to $3 \bmod 4$ is even. Then for large $N$ can we find $a$ and $b$, such that ...
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0answers
23 views

Sums of two squares theorem

Theorem: For each $n\in\mathbb{N}$, the number of integral solutions $x,y$ to the equation $n = x^2 + y^2$ is given by $4\sum_{d|n} \chi_1 (d)$, where $\chi_1$ is the Dirichlet character: $$ ...
6
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1answer
232 views

Increasing sequence of divisors of a quadratic trinomial

This question is from a Russian contest, the 2011 Tuymaada Olympiad. It's the fourth question on day two for the problems at grade level 2. Let $P(n)$ be a quadratic trinomial with integer ...
2
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2answers
38 views

PNT for composites

Is it true that \begin{align} &c_n\sim n+\operatorname{li}(n)+\operatorname{li}(n)/\log (n),\\ \end{align} where $c_n$ is the $n$th composite number? Is a better estimate known?
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2answers
28 views

if d divides n then prove that fibonacci of d divides fibonacci of n

prove that if $d$ divides $n$ then prove that fibonacci of $d$ divides fibonacci of $n$. i have tried to write $F(n)$ as a multiple of $F(d)$ using the fact that $n = ad$ for some natural $a$ but got ...
3
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3answers
90 views

For a prime $p\ge 17$ is $\dfrac{p^2-1}{24}$ ever a prime?

It was indicated in the comments of this MO question that if $p\ge5$ is a prime then $24|p^2-1$. Indeed $p=6k\pm1$ and $p^2-1=36k^2\pm12k+1-1=12k(3k\pm1)$ and exactly one of $k$ and $3k\pm1$ is even. ...
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2answers
66 views

Find all values of $x,y,z$ positive integers such that $4^x+4^y+4^z$ is a perfect square

I have to solve the equation $$4^x+4^y+4^z=k^2$$ I posted my solution but i don't know if there are other solution. How can i demonstrate that this expression is a perfect square? Are there oter ...
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0answers
37 views

Show this indivisibility

I have to learn for an exam but I can't solve this problem. Let $s_1, \cdots, s_n$ be positive integers, $n> 1.$ $t$ is a multiple of the product $\displaystyle \prod_{i = 1}^{n}{s_i}$. Prove ...
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1answer
74 views

equation $x^4 + y^4 = z^4$

Diophantine equations that are insoluble in $\mathbb{Z}$ may become soluble in finite integral domains. Show that \begin{equation*} x^4 + y^4 = z^4 \end{equation*} is soluble (as a congruence) in ...
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1answer
44 views

question on two-square problem.

Let $A_1, A_2$ be two quadratic residues of ($4k + 3$)-prime $p$ that satifsy $0 < A_1 < A_2 < p$. Prove that $A_1 + A_2 \equiv 0 \pmod p$ is impossible. Illustrate this result with $p = ...
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0answers
36 views

Provide me notes on Riemann zeta function to boast my knowledge to use in Research on Analytical Number Theory

I need your help. I want to study the Riemann zeta function from the very basic level, its concepts, theorems, solved problems etc. I am assigned one problem from Analytical Number Theory related to ...
2
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0answers
39 views

Prove an inequality of composite numbers

Yesterday after reading this post I tried to prove the inequality as given in the post. The inequality is, $$c_m+c_n>c_{m+n}$$ for all $m,n\ge1$. The problem was regarding the following special ...
2
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3answers
54 views

Why is it that if you square two prime numbers and add them, you get a number that is even and is not a perfect square?

If you do $x^2 + y^2 = n$ where $x$ and $y$ are both prime numbers and are both greater than $3$, why is $n$ always an even number that isn't a perfect square?
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3answers
37 views

When $p=3 \pmod 4$, show that $a^{(p+1)/4} \pmod p$ is a square root of $a$

Let $a$ and $p$ be integers such that $p$ is prime, and $a$ is a square modulo $p$. When $p\equiv3\pmod4$, show that $a^{(p+1)/4}\pmod p$ is a square root of $a$. Why does this technique not work when ...
3
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1answer
45 views

Could all iterates of $s(n)=2n+1$ be composite for some starting $n$?

Let $s(n)=2n+1$ and $\sigma(n)=\{n,s(n),s^2(n),s^3(n),\ldots\}$, where $s^3$ denotes functions composition, $s^3(n)=s(s(s(n)))$. For example $\sigma(11)=\{11,23,47,95,\ldots\}$. As another example ...
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0answers
35 views

what are the benefits of a factorial number system?

After reading an article about factorial number system. It tells that you can present any number in a factorial system and in if you have a number $a_{n-1}...a_2a_1a_0$ in factorial number system, you ...
0
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1answer
43 views

Form of $k$ when both $6k-1$ and $6k+1$ are primes

After a quick glance at sequence A007693 it seems that the following is true: if $p$ and $p+2$ are prime, then $\frac{p+1}{6}$ is prime. Questions: a) Is it the case? If not, what is the ...
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1answer
32 views

Finding a non-periodic sequence with prescribed complexity.

Let $\mathbf{a}$ be a sequence on a finite alphabet (i.e. set) $\mathcal{A}$. The complexity function of $\mathbf{a}$ is the number $p_{\mathbf{a}}(n)$ of distinct blocks of $n$ symbols (of ...
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1answer
79 views

30th problem of the fifth book of Diophantus;

Is there a complete answer to this problem? I have found Saunderson's answer, but I believe it is missing a few answers. The problem states: $a^2+b^2=d^2 \\ a^2+c^2=e^2 \\ b^2+c^2=f^2$ Saunderson ...
2
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0answers
40 views

Prove that the highest power of $n$ contained in $(n^r-1)!$ is $\frac{n^r-nr+r-1}{n-1}$.

Prove that the highest power of $n$ contained in $(n^r-1)!$ is $\frac{n^r-nr+r-1}{n-1}$. Attempt: I want to use the following theorem: The largest exponent of $e$ of a prime $p$ such ...
3
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4answers
67 views

Prove that number of zeros at the right end of the integer $(5^{25}-1)!$ is $\frac{5^{25}-101}{4}.$

Prove that number of zeros at the right end of the integer $(5^{25}-1)!$ is $\frac{5^{25}-101}{4}.$ Attempt: I want to use the following theorem: The largest exponent of $e$ of a prime $p$ ...
2
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1answer
12 views

W/ generating functions, How many solutions are there to the equation $2a+3b+c=n$ for some integer $n \geq 0$ and $a, b, c \geq 0$?

The question is: How many solutions are there to the equation $2a+3b+c=n$ for some integer $n \geq 0$ and $a, b, c \geq 0$? Solve this by writing down the correct generating function. I have no idea ...
2
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1answer
47 views

Is the sum of the reciprocals of the squarefree numbers divergent or convergent?

I just come across this question by trying to analyze the pseudoinverse of some infinite matrix (the matrix T as interpreted in my answer to this MSE-question), where this series occurs from some ...
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0answers
6 views

Unique symmetric multilinear form associated to a form

Let $F(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a form of degree $d$. In an article I am reading, it says to associate to $F$ the unique symmetric multilinear form $F(\mathbf{x}_1| ... | ...
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3answers
89 views

Show that for every prime $p$, there is an integer $n$ such that $2^{n}+3^{n}+6^{n}-1$ is divisible by $p$.

So the problem states: Show that for every prime $p$, there is an integer $n$ such that $2^{n}+3^{n}+6^{n}-1$ is divisible by $p$. I was thinking about trying to prove this using the corollary to ...
2
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0answers
41 views

Diophantine eqution with odd prime

HOW to find all possible set of solutions of an equation type $y^p \pm 2 = x^2$, where $p$ is any odd prime High regards to one and all
3
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1answer
68 views

Distribution of composite numbers

This question is moved from mathoverflow, there are several excellent answers at mathoverflow which improve my question greatly. For more information, please see the original question posted on ...
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0answers
38 views

Proof for primality based on exponents and primitive roots

I'm trying to prove the following statement: Suppose $n > 1$. The number $n$ is prime if and only if there exists a number $b$ such that gcd$(b, n) = 1$ and $b^{n−1} ≡ 1$ (mod $n$) and ...
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0answers
28 views

Largest known multi-perfect number (excluding perfect numbers)

What is the largest known multi-perfect number (excluding the perfect numbers) ? [2, 94; 3, 32; 5, 9; 7, 11; 11, 2; 13, 8; 17, 1; 19, 5; 23, 1; 29, 2; 31, 2; 37, 1; 43, 1; 53, 1; 59, 1; 61, 2; 67, ...
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0answers
60 views

Determine all natural numbers n and m that satisfying in this equation. [closed]

I'm trying to solve the following question: Determine all natural numbers n and m such that: $$n ^ { n ^ n } = m^m.$$ I don't have any idea about this question. Can somebody help me or give ...
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1answer
85 views

Quadruple of Pythagorean triples with same area

Can one find explicitly $a_i,b_i,c_i\in\Bbb N,i=1,2,3,4$ so that $$ a_i<b_i, \qquad \text{ and } \qquad a_i^2+b_i^2=c^2_i \qquad\text{for } i=1,2,3,4$$ and $$a_1b_1=a_2b_2=a_3b_3=a_4b_4, ...
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4answers
73 views

Let $a,b$ be relative integers such that $2a+3b$ is divisible by $11$. Prove that $a^2-5b^2$ is also divisible by $11$.

The divisibility for $11$ of $a^2 - 5b^2$ can be easily verified; in fact: $$a \equiv \frac {-3}{2}b \pmod {11}$$ therefore $$\frac {9}{4}\cdot b^2 - 5b^2 = 11(-\frac{b^2}{4}) \equiv 0 \pmod {11}.$$ ...
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0answers
63 views

Show that for a given $s$ there are a finite number of Fibonacci number of form $n^2+s$

It is well known that the last Fibonacci number $F_k$ such that $\exists \ n \in \Bbb{N} : F_k = n^2$ is $144$. Thus there are only $4$ perfect squares among the Fibonacci sequence (assuming you ...
12
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1answer
103 views

$CL(O_S) \cong \mathbb{Z}/3\mathbb{Z}$.

Let $F = \mathbb{Q}(T)$ and let $X$ be the set of all places of $F$, and let $S = \{w\} \subset X$ where $w$ is the place of $F$ corresponding to the maximal ideal $(T^3 - 2)$ of $\mathbb{Q}[T]$. Let ...
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27 views

Can this equation be solved given integer $n$ and constraint on $k$ and $e$ to be rational? $2k (e^2-n) + k^2 = 2n e^2 - 2n^2$

I am looking for a general way to solve for rational $e,k$ given integer $n$ $$2k (e^2-n) + k^2 = 2n e^2 - 2n^2$$ Repeating fraction methods are fine I just need a rational number for $k$ and ...
1
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1answer
25 views

Lower and upper bounds for $\tau(n)$

How to prove the following statement: If $n$ is the product of k powers of primes, i.e. $n=\prod\limits^{k}_{i=1}p_i^{\alpha_i}$ then $\omega (n) = k$ and $\Omega=\sum\limits_{i=1}^{k}\alpha_i$ $$ ...
3
votes
2answers
46 views

How to prove this asymptotic formula?

How to prove this asymptotic formula? $$ \prod\limits_{p\leq x}\left(1+\frac{1}{p}\right) \sim \frac{6 e^C}{\pi^2}\log x $$ Where we multiply over all primes less than or equal to x. I have little ...
1
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0answers
23 views

Why $\int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn$?

I am reading the lecture notes. On page 5, formula (1.24) is $$ \int_{U} d(m n m^{-1}) = \int_{mUm^{-1}} dn = \omega^2_E(t_1) \int_U dn, $$ where $dn$ is the Haar measure on $N$, $U \subset N$ is ...
6
votes
2answers
79 views

${n}\choose {r}$ =$ 8$ Is there any way to find such $n$ and $r$?

Let ${n} \choose {r}$ = $8$. Is there any other choice of $n$ and $r$ except $8$ and $1$, $8$ and $7$ ? In general how to check that existence is guaranteed or not?
2
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1answer
59 views

Number $N>6$, such that $N-1$ and $N+1$ are primes and $N$ divides the sum of its divisors

The perfect number $6$ is in the middle of the primes $5$ and $7$. It is the only perfect number with this property because odd numbers are not in the middle of two twin primes and even perfect ...
8
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0answers
85 views

Remainder of dividing $3^n$ by $2^n$.

I'd like to find the remainder of dividing $3^n$ by $2^n$, that is, I'd like to find value of $r$ in the expression $$3^n=q2^n+r,$$ where $q\in\mathbb{Z}$ and $0<r<2^n$. I know that it can be ...
11
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0answers
74 views

Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$?

It is known that given primitive (co-prime) integer solutions to, $$x_1^4+x_2^4+x_3^4+x_4^4 = z^4$$ then there is one $x_i$ such that $z^4-x_i^4$ is divisible by $d_4=5^4$. Additionally, Ward ...