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 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
59 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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2answers
22 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
31 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 ...
2
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2answers
27 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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2answers
77 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 ...
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1answer
42 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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36 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
31 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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1answer
48 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
65 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 ...
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24 views

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

Let $n>1$ be an integer; we call $n$ a "ring number" if there exists a commutative ring $R$, with identity, having exactly $n$ ideals (including $\{0\}$ and $R$); now since for every $n>1$, ...
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1answer
33 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?
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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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1answer
65 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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71 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 ...
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1answer
29 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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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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3answers
61 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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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 $\mathbb Q$ ...
3
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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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1k 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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1answer
18 views

Half primes in the set

Let S be 30 element subset of {1,2,....2015} such that every pair of elements in S are relatively prime. Prove that at least half of the elements in S are prime numbers
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1answer
36 views

Number theory, prove that a prime number $p \mid 1$

Consider a prime number $p > 1$ and $a \in \mathbb{Z}$ and $p < a$. We know $p \mid a$, then $a = p.b$ for $b \in \mathbb{N}$. We also already know the congruence $a \equiv 1 (\text{mod } m)$ ...
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2answers
24 views

Proving that for every prime number $p$ there exist $a,b \in \mathbb{Z}$ such that $-1 \equiv a^2+b^2\ (mod\ p)$

How can I prove that for every prime number $p$ there exist $a,b \in \mathbb{Z}$ such that $-1 \equiv a^2+b^2\pmod p$?
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2answers
25 views

For which primes $p\not=2$ is $5$ a square mod $p$?

For which primes $p\not=2$ is $5$ a square mod $p$? Using the Legendre symbol, $5$ is a square modulo $p$ if $$\left(\frac{5}{p}\right)=5^{\dfrac{p-1}{2}} \equiv 1 \pmod{p}$$ Now we have ...
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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 ...
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0answers
13 views

Identity for exponential character sums

I was confused about the following identity I ran into. I would appreciate it if somebody could clear this up for me. Suppose that we have an exponential sum $$g(a)=\sum_{t=0}^{p-1} \exp( 2\pi i ...
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2answers
62 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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1answer
19 views

Number of ordered pairs with given lcm

Suppose $(p,q)$ is an ordered pair of natural numbers with lcm $r^2.s^4.t^2$, where $r,s,t$ are distinct primes.We have to find the number of all such ordered pairs. One simple idea I tried is to fix ...
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2answers
62 views

Prove that $a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$ for $a, b, c > 0$

Prove for $a, b, c > 0$ that $$a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$$ Could you give me some hints on this? I thought that Jensen's inequality might ...
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0answers
46 views

If $\gcd(a,b) = \gcd(c,d) = 1$ and $ab = cd$, then $a=c$ and $b=d$. [on hold]

Is this conjecture true? If yes, can somebody help me prove it? If not, can anyone come up with a counter example?
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0answers
28 views

Z is a subset of Q (the set of all rational numbers)

Let $$n \in Z$$ Then $$n*1=n$$ and so $$n=n/1$$ Note n and 1 are both in Z. so n can be written in the form of $$ z = m/n,\,\,\, where\,\, m,n \in Z\,\,\,and\,\,\, n ≠ 0$$so $$n\in Q$$ Is it enough ...
2
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2answers
84 views

Proof that there are infinitely many primes (Euclid)

I was wondering if I could get some insight on my proof. I am in the midst of relearning some number theory and just "writing proofs" in general, and I would like some assistance to see if I am on the ...
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1answer
28 views

Solve for bound of $\sigma(n)$ from harmonic series.

I am given the harmonic series, $H_n$. How can I show that for $n\geq 1$, $$\sigma(n)\leq H_n+e^{H_n}\ln{H_n}$$ By the way, if you're not familiar with it, $\sigma(n)$ is the sum of all positive ...
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3answers
52 views

Rational number proposition

**Prop.**Every $$r \in Q$$ can be written as r = m/n, where $$ m,n \in Z$$ such that n>0 and gcd(m,n) = 1 (r is in lowest terms) If I start by saying that let $$r \in Q$$ Then there exist $$a,b \in ...
2
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1answer
25 views

Classification of moduli where relatively prime numbers squared are 1

I came across an interesting property of certain numbers with respect to modular arithmetic and I was wondering if anybody had any more information about them. Consider an integer $n$ such that if ...
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1answer
25 views

Conditions for existence of quadratic residue congruent to 1

Under what conditions are we guaranteed an existence of quadratic residue 1 other than squares of 1 and -1. What conditions a number must satisfy to have such residue.
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0answers
9 views

Finding the roots with the largest magnitude

Given a non-constant polynomial $p\in\mathbb{Z}[x]=\alpha\prod_{k=1}^nx-\alpha_k$ how can I find the roots $R=\{\beta_1,\ldots,\beta_t\}\subseteq\{\alpha_1,\ldots,\alpha_n\}\subseteq\mathbb{C}$ with ...
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1answer
94 views

How many divisors does $111…1$ have?

Let $A=\underbrace{11..1}_{2010}$. How many divisors does $111...1$ have? Original problem: Prove that $τ(A)>50$ (or $τ(A)<50$) My work so far: If $\tau(A) -$ the number of divisors ...
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1answer
19 views

Multiplicative Inverse Modulo N.

I would like to be able to figure out the multiplicative inverse of some integer modulo some N. For example, how would I find ${15^{ - 1}}$ modulo 34. Will these always exist? If not, what dictates ...
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0answers
39 views

Find the value of $n$

Let $F$ be a field having $5^n$ elements .Also $F$ has an element which satisfies $x^{5^n}=1$ such that $x\neq 1$. Find $n$ . My try: Let $x\in F $ satisfy $x^{5^n}=1$ .Obviously the group ...
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1answer
148 views

Solving a Word Problem relating to factorisation [closed]

The $\text{Ionof}$ (Integer on number of factors) of an integer is the integer divided by the number of factors it has. For example, $18$ has $6$ factors so $\text{Ionof}(18) = \frac{18}{6} = 3$, and ...
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1answer
34 views

How to find scaling to get minimum positive integer proportion?

Suppose we have x is a strictly positive vector and y=b*x where b is a positive scaling scalar. The problem is to find the function to get the scaling factor b such that y becomes minimum positive ...
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2answers
47 views

arithmetic mean of smallest numbers of all subsets of r elements formed out of (1,2,..n)

Consider all subsets of r elements of the set $\{1,2,3,......,n\}$ where $1 \leq r \leq n$. Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest ...
3
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1answer
112 views

How many pairs $ (a,b)$ of integers such that , $a^2b^2=4a^5+b^3 $

I would appreciate if somebody could help me with the following problem: $Q$: How many pairs $ (a,b)$ of integers such that $$a^2b^2=4a^5+b^3 $$
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1answer
25 views

What is the bank need to get the message?

In Number theory $p=37, q= 43$, $\phi(pq)= 36 \cdot 42$, $e=5$ $d=?$ What does the bank need to get the message? I don't understand this problem. Can any one help me please?
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0answers
25 views

Generating functions, Schur's identity

Let $S=\{n\in \mathbb{Z}_+ \mid n \equiv 1, 5 \,\,(\text{mod 6})\}.$ Let $a(n)$ be the number of partitions of $n$ into parts belonging to $S,$ and $b(n)$ be the number of partitions of $n$ into ...
2
votes
1answer
26 views

How to find a square root mod $pq$ given that $p \equiv q \equiv 3 \pmod 4$

Let $n = pq$ where $p$ and $q$ are prime. We do not know $p$ and $q$. All we know is that $p \equiv q \equiv 3 \pmod 4$. From this we need to find a number $y$, in terms of $n$ and $x$, such that $y^2 ...