Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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128 views

Binary sequences in primes

Is anything known about these problems? If we make a string S of 0's and 1's with 1 in n'th position if the the nth prime $p_n$ is of the form $1+m 2^{9^{9^{9^{9}}}}$, else 0, does every finite string ...
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0answers
251 views

Easiest way to prove that a subset of even integers is closed under multiplication?

What's the easiest way of showing that; $2\mathbb{Z}\setminus (4n-2)\mathbb{Z}$ is closed under multiplication? (I'm trying to show that $(4n-2)$ is a prime element of $2\mathbb{Z}$ by showing ...
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177 views

Galois Group over Ring of Integers

Suppose we have a quadratic (Galois) extension of $\mathbb{Q}$, call it $k$ with Galois group $G$. If we look at the ring of integers inside of $k$, call it $\mathcal{O}_k$, is it true that ...
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92 views

Are the signs of these eigenvalues from this Hermitian matrix equal to the Möbius function?

I am partly repeating myself here. Are the signs of these eigenvalues from this Hermitian matrix "c" equal to the Möbius function? Eigen99 in the Mathematica code is the list of eigenvalues for a ...
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0answers
117 views

Does number theory have any role in the proof of convergence of Fourier series for certain functions?

Does number theory have any role in the proof of convergence of Fourier series for certain functions? I vaguely remember reading in a book on signal processing, way back, that the proof (original ...
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1k views

Teach me a simple, efficient division algorithm

I want to implement arbitrary-precision arithmetic in JavaScript for non-negative integer numbers. Long division isn't efficient if instead of the usual 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) there ...
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0answers
215 views

Dirichlet's Class Number and its connections with the $GL(2)$

i posted the same question on MO,but cant get an answer so i am trying here note:all those who answer my question just mention the question number in their reply so that i can tally them,thanks a ...
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0answers
65 views

Can Fermats descent be interpreted on a conic?

Fermat proved the Diophantine equation $$(x^2)^2 + (y^2)^2 = z^2$$ has only solutions $(0,0,0)$, $(0,\pm 1,\pm 1)$ and $(\pm 1,0,\pm 1)$ using "infinite descent". The conic $C : X^2 + Y^2 - 1$ has a ...
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135 views

Number theory - Divisibility and perfect square related [duplicate]

Possible Duplicate: Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer. I've tried many different approaches and they don't seem to lead anywhere. The question ...
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281 views

Number of solutions to an equation using the inclusion-exclusion principle [duplicate]

Possible Duplicate: Inclusion-exclusion principle: Number of integer solutions to equations This is a problem that I have had much trouble with. If you attempt in helping me with this, ...
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0answers
212 views

$(a^n +b^n)/((ab)^{n-1}+1)$ is a perfect $n^{th}$ power

Let $a,b$ be positive integers satisfying $$(ab)^{n-1}+1 \mid a^n +b^n.$$ Then how to show that the number $\frac{a^n +b^n}{(ab)^{n-1}+1}$ is a perfect $n^{th}$ power of an integer? Another question: ...
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111 views

Potential computational questions that could be asked about p-adic numbers and Galois Theory

I have an exam on P-Adic integers (and a bit on Galois Theory) that my professor said would be very computational, but he never does any examples of the theorems he proves in class. He said the exam ...
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0answers
79 views

Are limits on exponents in moduli possible, if the modulus is relatively prime?

I asked a similar question to this recently. Here, I consider an arbitrary, but fixed, modulus m, which is relatively prime to x and y. Can anybody extend the answer given in the previous question? ...
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0answers
110 views

Is that series-transformation known in the context of divergent summation

Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the ...
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1answer
102 views

Divide N Hot dogs among M persons

There are N hot dogs and M people and we need to divide the hot dogs equally. Now we need to calculate the minimum number of cuts required to distribute the hot dogs equally. In order to divide the ...
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2answers
39 views

Is this a good expression of the claim using logical notation?

Four Square Theorem: Every positive integer can be written as a sum of four integer squares. Expressed in logical notation : $$\forall n>0 =a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2}$$
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1answer
172 views

Quadratic residues for odd primes

Problem: For each odd prime $p$, we consider the two numbers $A=$ sum of all $1 \leq a < p$ such that $a$ is a quadratic residue modulo $p$, $B=$ sum of all $1 \leq a < p$ such that $a$ is a ...
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1answer
91 views

How to minimize the difference of two factors when prime factorization is available

There are nonzero natural numbers ($\geq 1$) $a,b,c,d$. $c$ is fixed, and prime factorization of $c$ is available. The prime factorization of $c$ always have the same nonzero exponent - that is ...
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1answer
91 views

How prove there exist prime numbers $P_{1},P_{2},\cdots,P_{n}$ such $P_{k}\mid c+k$

Question: Let $c,n\in\mathbb{N}$, such that, $c>n^{n-1}$. Show that: there exist distinct prime numbers $P_{1},P_{2},\dots,P_{n}$ such that: $$P_{k}\mid c+k,k=1,2,3,\dots,n$$ My ...
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2answers
109 views

Two positive integer with prime number

Let $a, b$ be distinct positive integers. Prove that there exists a prime $p$ such that when dividing both $a$ and $b$ by $p$, the remainder of $a$ is less than the remainder of $b$. How can i solve ...
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7answers
108 views

$(a,b) = 1$ implies $a|n$ and $b|n \implies ab|n$

Prove if $(a,b) = 1$ implies $a|n$ and $b|n \implies ab|n$. I'm pretty sure this has been asked before but I cannot find anything online.... I also have no idea how to solve it, I get stuck with al = ...
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4answers
355 views

Solving $\phi(n)=22$

Find all positive integers n such that $\phi(n) = 22$ and prove that there are no others. Here $\phi$ denotes the Euler $\phi$-function.
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3answers
206 views

$ax^3+by^3+cz^3=0$ and Elliptic curves

What is relation between $ax^3+by^3+cz^3=0$ and Elliptic curves?
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2answers
163 views
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3answers
418 views

About some Pythagorean Quadruples

I am trying to find all the Pythagorean Quadruples of the form: $$ 1+(10K+4)^2+(10M+8)^2=(10N+9)^2\qquad K,M,N\in\mathbb{N},M<N $$ Thank you!
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3answers
100 views

Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares

I'd appreciate your help showing that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then p can't be written as a sum of two squares. Thanks!
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1answer
384 views

Resurrection of my Tamagawa numbers Question, to understand the Formulation of BSD

My previous question was closed very badly for asking the broad and deep things, so I now understand the consequences of asking such questions, so I refrain from asking such questions, so this is not ...
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2answers
193 views

Proving if $f(x) \in \mathbb{Z}[x]$ and $f(a) \cdot f(b) = -(a-b)^{2}\neq 0$ then $f(a)+f(b)=0$

I am having trouble solving this problem. Suppose $f(x) \in \mathbb{Z}[x]$, and $f(a) \cdot f(b) = -(a-b)^{2}$ for distinct $a,b \in \mathbb{Z}$, then how do we prove that $f(a)+f(b)=0$? Don't ...
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3answers
255 views

Congruences of the form $x^2-a \equiv 0$ (mod pq)

Problem: Let p and q be distinct primes. What is the maximum number of possible solutions to a congruence of the form $x^2-a \equiv 0$ (mod pq), where as usual we are only interested in solutions that ...
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1answer
146 views

Prove that $p\mid \binom{p}{k},\ 0< k< p$

Prove that: $$p \,\,\left|\, {p \choose k} \right., \quad 0< k \lt p$$ if $p$ is prime. how to prove that with direct proof?
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2answers
2k views

Sum of the factors of any natural number

Prove that the sum of all the factors of any natural number is equal to $\frac{a^p+1 -1}{a-1}$ [YF] Here is one possible interpretation of the above: Prove the formula for the sum-of-divisors ...
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2answers
129 views

How to find the roots of polynomials in $\Bbb Z_p$

Consider the polynomial $x^{p-1} - 1$ in $\mathbb{Z}_p$. What are its roots / how does it factor? Does this factorization tell you anything about $(p-1)!$ modulo $p$? I'm really stuck on this ...
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3answers
266 views

GCD (p+q,p-q) with distinct odd primes

Suppose $p$ and $q$ are distinct odd primes. Prove that $\gcd(p+q, p-q) = 2$.
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4answers
796 views

$\sqrt{17}$ is irrational: the Well-ordering Principle

Prove that $\sqrt{17}$ is irrational by using the Well-ordering property of the natural numbers. I've been trying to figure out how to go about doing this but I haven't been able to.
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5answers
100 views

Finding the largest $n \in \mathbb{N}$ for which $n-7$ divides $n^3-7$

I want to find the largest $n \in \mathbb{N}$ for which $n-7$ divides $n^3-7$. In other words, I am looking for the largest $n$ such that $\frac{n^3-7}{n-7}$ is an integer. Can anyone provide me with ...
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3answers
153 views

Finding the number of integer solutions, why is this wrong?

The question is to find the number of solutions such that $(x, y)$ are integers: $(x-8)(x-10)=2^y$. Here's what I did: $u(u-2)=2^y$. From the quadratic formula, $u=1+\sqrt{1+2^y}$. This is where I ...
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3answers
220 views

a basic question on relatively prime numbers

Let $p,q$ belong to $\mathbb{N}$ and are relatively prime to each other. If $\alpha,\beta$ belong to $\mathbb{N}$, are also relatively prime to each other,then are $(p\beta+q\alpha)$ and $q\beta$ ...
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1answer
138 views

There is no solution for this equation [closed]

Prove that $3^a - 2^a \equiv 0 \pmod a$ for a natural number greater $a>1$ , has no solution .
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1answer
81 views

Finding positive integer solutions to $3^x + 55=y^2$

I think it must be finite, $y$ is always even, but I don't know how to continue. edit: with $x,y\in\mathbb Z$
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4answers
154 views

$(a,b)=d \overset{?}{\implies} (a^3,b^3)=d^3$

Why is this true? I suspect that its because $\frac{LCM(a,b)^3GCD(a,b)^3}{b^3}=a^3$ and $\frac{LCM(a,b)^3GCD(a,b)^3}{a^3}=b^3$, so it must be the case for $LCM(a,b) \notin R(a,b)$, right?
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2answers
80 views

Correct $(0.11111111…)_{2}=(10)_{2}$?

My question is if it is possible to claim that $(0.11111111...)_{2}=(10)_{2}$ Here is my approach. I started out by trying to convert this recurring number to base 10. This can be expressed as a sum, ...
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4answers
144 views

number theory $\gcd(a,bc)=\gcd(a,c)$

Suppose $\forall (a;b;c;d) \in \mathbb{Z}^4$ as $\gcd(a,b)=\gcd(c,d)=1$: How can I prove that $\gcd(a,bc)=\gcd(a,c)$? Also, how to prove that $\gcd(ac,bd) = (\gcd(a,d))*(\gcd(b,c))$?
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3answers
228 views

find all rational numbers p/q such that $|p/q-17/5|< 1/q^2$

This is a question in my assingment. I needto find all rational numbers p/q such that $|p/q-17/5|< 1/q^2$. Any ideas ? Thanks for any help!
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2answers
162 views

Mordell equation implemented anywhere?

I admit I have no idea how to tag this post, but I'm looking for a CAS/number theory software package that would implement a decent algorithm for computing the integral solutions to $x^2 = y^3 - k$, ...
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2answers
197 views

The Diophantine equations $X^3=DY^3+A^3$

Does anyone know for which values of $D$ the equation $X^3=DY^3+A^3$ has solutions? All numbers non-zero naturals.
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3answers
341 views

Do there exist any dormant primes?

Suppose that n is an integer > 1 such that: The prime factorization of n is known It is known that (n + 1) is a prime Then: What can be concluded? Among the possibilities are the following: We ...
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4answers
58 views

Phenomenon regarding square of any integer…

There is a phenomenon regarding squares of integers which i observed today. $n^2 = \sum_1^n^-^1 + \sum_1^n $ I am a computer science graduate and i never heard about this phenomenon till date. Is it ...
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3answers
90 views

Algorithm for Fundamental theorem of arithmetic

What is a good algorithm for decomposing a number into a product of primes? What would be its time complexity?
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5answers
48 views

What are the possible values of the last digit of $4^m$, where m is a natural number?

What are the possible values of the last digit of $4^m$, where m is a natural number? After trying a couple of m=1,2,..., I found that the possible values for the last digit are 4 if m is odd and 6 ...
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2answers
87 views

Is $x^4+x+1$ irreducible in $\Bbb{Q}[x]$?

Decide with a proof if $f(x)=x^4+x+1$ is irreducible in $\Bbb{Q}[x]$. ** End Question ** I was thinking of using DeMoivre's Theorem but I'm not sure how. Thanks!