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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Find the simple continued fractions for both $\pm \frac{39}{25}$…

Find the simple continued fractions for both $\pm \frac{39}{25}$? So far for $\frac{39}{25}$ I have: $39 = 1 \times 25 + 14 $ $ 25 = 1\times 14 + 11 $ $14 = 1 \times 11 + 3$ $11 = 3 \times 3 ...
3
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0answers
25 views

Can we find prime numbers with any sum of digits (except those divisible by three)

I guess that this question is not something new and that there must be people who wanted to know if this question has an affirmative answer, but I would like to share it with you, because I really do ...
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2answers
37 views

if $a_1 \cdot b \equiv a_2 \cdot b (\text{mod }n)$ then $a_1 = a_2$

Assume $a_1 \cdot b \equiv a_2 \cdot b (\text{mod }n)$ and also $a_i^{\frac{n-1}{2}}$ mod $n \in \{1, n-1\}$ and $b^{\frac{n-1}{2}}$ mod $n \notin \{1, n-1\}$, I saw the following which proves $a_1 = ...
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1answer
17 views

Isomorphism of completions of number fields

Let $K$ and $L$ be number fields, $v$ a place of $K$ (either archimedean or non-archimedean) and $\theta:K\simeq L$ a ring isomorphism. I am trying to show that $\theta$ induces an isomorphism ...
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0answers
15 views

Number of iterations of an “integer-logarithm”

Let us consider the function $\sigma:\mathbb{N}\to\mathbb{N}$ defined as: If $\prod_{i=1}^{r}p_i^{\alpha_i}$ is the prime-factorization of $n$, then $$ \sigma(n)=\sum_{i=1}^{r}\alpha_ip_i $$ So in ...
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0answers
24 views

For a prime p and a positive integer n

we define $A_{p,n} = \{(x,r) : 1 \leq x \leq n \textrm{, r is a positive integer, } p^{r} \textrm{divides x} \}$. Describe the set $A_{p,n}$ for p=5 , n=100. Does the set comprise of ...
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3answers
28 views

Finding Maximum Mod

Given a set of numbers, say $x=\{1,2,3\}$, how can I find the maximum $m$ such that $x_i\bmod m =x_j\bmod m$, where $i$ and $j$ are some indexes of the set $x$. So for $x=\{1,2,3\}$, the answer should ...
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1answer
54 views

If $d>1$ is a squarefree integer, show that $x^2 - dy^2 = c$ gives some bounds in terms of a fundamental solution.

If $d>1$ is a squarefree integer, show that $x^2 - dy^2 = c$ gives some bounds in terms of a fundamental solution. I am not able to understand the question itself. What does it exactly mean ...
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3answers
136 views

How many values does the expression $1 \pm 2 \pm 3 \pm \cdots \pm n$ take?

How many different values does the expression $1 \pm 2 \pm 3 \pm \cdots \pm n$ take? I was wondering about this problem and didn't think it was immediately obvious. The answer can't be $2^{n-1}$ ...
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127 views
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Find a closed form formula for $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes.

I meant by "closed form formula" a formulate that doesn't have summation or has very few terms. Maybe there's a better term for this meaning. I found this function that has very interesting property ...
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1answer
29 views

Looking for a simpler solution about quadratic congruence

Here is the Problem: 1)Suppose $p$ is a prime. prove that for any integer $k$, there exist integers $x$ and $y$ such that $x^2+y^2 \equiv k\ \pmod p$. 2)Are there infinitely many composite ...
4
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2answers
113 views
+50

Find all integral solutions for the Diophantine Equations $x^4 - x^2y^2 + y^4 = z^2$ and $x^4 + x^2y^2 + y^4 = z^2$.

Find all integral solutions for the Diophantine Equations $$x^4 - x^2y^2 + y^4 = z^2$$ and $$x^4 + x^2y^2 + y^4 = z^2$$ I basically think that to solve these equations we need to use the fact ...
3
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3answers
29 views

Does the PNT establish a connection between primes and the logarithm?

The prime number theorem states that $$\pi(x) \sim \frac x {\ln(x)}$$ Morally, this seems to suggest that there is a fundamental connection between primes and the natural logarithm. But since we're ...
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0answers
14 views

Show that points on an elliptic curve have order 4

I am studying elliptic curves using this book and have a problem with task 4.11 which goes as follows: Let $F_q$ be a finite field of odd characteristic and let $ a,b \in F_q $ with $a \ne2b$ and $b ...
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3answers
53 views

Alternative Proof: if $n$ is an integer, prove that $\frac{n ( n^4 - 1)}{5}$ is an integer

I have proven this by the induction method but would like to know if it can be proven using an alternative method.
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4answers
5k views

Primes of the form $n^2+1$ - hard?

I met a student that is trying to prove for fun that there are infinitely many primes of the form $n^2+1$. I tried to tell him it's a hard problem, but I lack references. Is there a paper/book ...
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2answers
133 views

Let $d$ be any positive integer not equal to $2, 5,$ or $ 13$ , then $\exists a, b \in \{2, 5, 13, d\}$ such that $ab − 1$ is not a perfect square?

Let $d$ be any positive integer not equal to $2, 5,$ or $ 13$. Then can we always find distinct $a, b \in \{2, 5, 13, d\}$ such that $ab − 1$ is not a perfect square ?
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0answers
58 views

“Matrix sieve” - primes finding algorithm - is it useful for number theory? [on hold]

I proposed "matrix sieve" algorithm for finding primes that in my opinion is simple, not needed operations of dividing and easy to memorize: In order to find all primes (up to a given limit) in the ...
3
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2answers
302 views

Infinitely many primes in the ring of integers

Let $K$ be a number field such that $\mathcal{O}_K= \mathbb{Z}[\alpha]$ for some $\alpha$ algebraic integer. Prove that there are infinitely many primes $\mathcal{P} \subset \mathcal{O}_K$, such ...
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2answers
61 views

$a,b$ are positive integers . Prove that if $k=\frac{a^2+b^2}{ab-1}$ is an integer then $k=5$. [on hold]

$a,b$ are positive integers . Prove that if $k=\frac{a^2+b^2}{ab-1}$ is an integer then $k=5$. I tried to see the whole expression as a quadratic of $a$ but that is not helping much
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2answers
18 views

Number Bases and quadratic equations

could anybody help me with my maths extension assignment? This is the whole question
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1answer
30 views

what is a good book (about number theory) [duplicate]

I find a good books for number theory... An Introduction to the Theory of Numbers (by G.H hardy)or Burton, Rosen..etc is it good?? (i want a book which best of best in number theory) i want a lot ...
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1answer
28 views

Denominators of harmonic numbers: asymptotic behaviour.

About the sequence $d_n$ of the denominators of harmonic numbers, I know these facts: It is unbounded, since $p\mid d_p$ for any prime $p$. It contains only one $1$. What more is known? Specially, ...
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2answers
26 views

Charmichael number square free

Show that if $n$ is a Charmichael number, then $n$ is a square-free. I did this: Let $n= (p^t)(m)$ where $t >1$. Then by modular property, $$b^p= b \mod n , \,\, b^m= b \mod n$$ Above two ...
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0answers
101 views

Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
2
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1answer
36 views

Do local Galois representations always lift?

Suppose $G:=G_F$ is the absolute Galois group of a local (residue char. $\ell$) or global field $F$, and $\bar{\rho}$ a (linear) representation of $G$ on the $\mathbb{F}_q$-module $\mathbb{F}^d_q$, a ...
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2answers
64 views

Without calculating them determine whether $36^2+1$ and $154^2+1$ are prime and find the prime factors if not prime

I know that $36^2 + 1$ is prime, $154^2 + 1$ is not, both are equal to $1 \bmod 4$. The prime divisors of $154^2 + 1$ should also be of the form $1 \bmod 4$. Tried showing this by Wilson's theorem ...
2
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1answer
30 views

Is it always true, for a prime $p$, a generator $g$ of $\mathbb{Z}^*_p$ cannot be a quadratic residue modulo $p$?

Let $p$ be a prime, then is it true that a generator $g$ of $\mathbb{Z}^*_p$ cannot be a quadratic residue modulo $p$? And if yes why?
3
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2answers
50 views

Finding the sum of all products of pairs of distinct primitive roots mod 83

I'm currently studying Number Theory and I've stumbled upon a question where I need to: Find the sum of all products of pairs of distinct primitive roots mod 83. Solving attempt: I've tried to find ...
5
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2answers
135 views

Maps of primitive vectors and Conway's river, has anyone built this in SAGE?

I am attempting to teach number theory from John Stillwell's Elements of Number Theory in the upcoming semester. There are two sections (5.7 and 5.8) which describe the diagrammatic method for the ...
6
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1answer
129 views

Why can this cosine sum function show all primes less than $N^2$?

I constructed this cosine sum that puts all primes within N on line y=1, and its zeros show the sieve by primes less than N. For $x<N^2$, they are all primes. $$ ...
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2answers
40 views

Totien-Sum: why GCD( {n}/d, q/d) = 1; implies Sum{Totient(d/q) } = q

Have seen answer to this question. still don't understand.. Totient sum is defined: q = Sum(Totient (d) ); sum on all d : d|q More specific; The proof has these steps: 1. If d is a divider ...
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1answer
60 views

The number of integral solutions $(x,y)$ of $x^3+3x^2y+3xy^2+2y^3=50653$

This was a wonderful question given to me by professor in my last class test. He asked for the solution with the least number of steps. Find the number of integral solutions $(x,y)$ of the ...
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1answer
39 views

Why is the absolute value of this Gauss sum obvious?

I came across the Gauss sum discussed in the following post in a problem from my Galois theory course: http://mathoverflow.net/a/71282. Why exactly is the square of its norm obvious?
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0answers
966 views

Integer Triangle Radicals conjecture

An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the ...
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0answers
33 views

$(k^2)! \cdot \prod_{j = 0}^{k = 1} {{j!}\over{(j + k)!}}$ integer for $k \in \mathbb{N}$

How do I see that for any positive integer $k$,$$(k^2)! \cdot \prod_{j = 0}^{k = 1} {{j!}\over{(j + k)!}}$$is an integer?
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0answers
30 views

proof of number of sub arrays of an array of size $N$ using combinatorics

What is the proof that number of sub arrays of an array of size $N$ is $$\frac{N(N+1)}{2}$$
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2answers
18 views

On those integers $n>1$ such that there exist a coommutative ring with identity with exactly $n$ ideals

Let $n>1$ be an integer ; we call $n$ a " ring number " if there exist a commutative ring $R$ , with identity , having exactly $n$ ideals ( including $\{0\}$ and $R$ ) ; now since for every ...
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67 views

Let $xy|(x^2+y^2+1)$. Prove that $\frac{x^2+y^2+1}{xy}=3$ [duplicate]

Let $x,y -$ positive integers, such that $xy|(x^2+y^2+1)$. Prove that $$\frac{x^2+y^2+1}{xy}=3.$$ My work so far: 1) If $x=y=1$ then $\frac{1^2+1^2+1}{1}=3$ 2) Let $x=1$ (or $y=1$) ...
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0answers
44 views

Functional equation in natural numbers $x+y|f(x)+f(y)$

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $x+y|f(x)+f(y)$ and for $x\ge1395$ we have $2f(x)\le x^3$. What I've tried so far: For $x=y$ we get $2x|2f(x)$ and $x|f(x)$. It's ...
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3answers
122 views

System of congruences and Chinese remainder theorem

Find all the integers satisfying this system of congruences $$\begin{cases} x \equiv 2 \pmod 5\\ x \equiv 1 \pmod {10}\\ x \equiv 0 \pmod 3 \end{cases} $$ I think you use Chinese remainder theorem ...
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1answer
28 views

The Archimedean place of $\mathbb{Q}$

Is there a way to extract the Archimedean absolute value of $\mathbb{Q}$ from its field structure in a way analogous to its non-archimedean absolute values? Here is some context: Given a valuation ...
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1answer
33 views

There exists an irrational number z such that x<z<y

I know there are lots of post about this but I wanted to know this proof would work also. Proposition. Let $x,y ∈ \mathbb{R}$ with $x < y$. There exists an irrational number $z$ such that $x < ...
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1answer
27 views

How to prove that if the sum of the totatives of two numbers is equal then the numbers are equal?

As the title says, I am trying to prove that if the sum of the totatives of $a$ equals the sum of the totatives of $b$ then $a = b$ but I am stuck. I have that sum of totatives of $n = f(n)= ...
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0answers
21 views

Does a place $v$ of a number field $K$ ramify in $L/K$ iff $v\mid d_L$?

Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$. I know that a rational prime $p$ in $\mathbb Q$ ...
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0answers
28 views

The Divisors of $s(2s+1)$ and Primes $n$, $4n+1$, and $6n+1$

This question is somewhat related to this one. Most of this is by way of a computer search: claim: If $s$ is any positive integer I write $\varphi_{s} =s(2s+1)$. Let $\tau$ be the divisor ...
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58 views

Pell's equation and binary hyperbolic forms.

We define the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $0\neq c=a^2+b^2$. Is it true that $f$ is hyperbolic? In other word, is there any ...
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0answers
27 views

If sum equals one, then at least two terms are relatively prime

Suppose that we have some set $A = (a_i)_{i =1}^k$ of integers and that $$ \sum_{i = 1}^k c_i a_i = 1 $$ for some integer constants $(c_i)_{i=1}^k$. Does this imply that $\gcd(a_i,a_j)= 1$ for some ...
3
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3answers
58 views

Let $A = (0,1]$. Then$\text{ inf}(A) = 0$

I posted before about this proposition and I thought I got it right but then I was told that it is still wrong so I am really confused again.. Here is my proof Proof : Let $A = (0,1]$ Here, since ...
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1answer
21 views

Every positive integers of the form $4k+1$ can be factored into Hilbert primes

How can I show that every positive integer of the form $4k+1$ can be factored into Hilbert primes? A Hilbert prime is defined as a positive integer of the form $4k+1$ without a smaller factor of this ...