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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0
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0answers
20 views

How prove this $3^{\frac{5^{2^n}-1}{2^{n+2}}}\equiv (-5)^{\frac{3^{2^n}-1}{2^{n+2}}}\pmod {2^{n+4}}$

Question: show that: $$3^{\frac{5^{2^n}-1}{2^{n+2}}}\equiv (-5)^{\frac{3^{2^n}-1}{2^{n+2}}}\pmod {2^{n+4}}$$ My idea: since I have prove $$5^{2^n}-1\equiv 0\pmod {2^{n+2}}$$ ...
0
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1answer
39 views

fastest algorithm for prime factorization [on hold]

I need the fastest algorithm to factorize the given number $N$ as a product of primes. $$N=p_1^{e_1}p_2{e_2}\ldots p_n^{e_n}$$ where $p_1, p_2,\ldots ,p_n$ are primes and $e_1,e_2,\ldots, e_n$ are ...
1
vote
2answers
26 views

Help with proof regarding degrees of polynomials

How do you prove that if $f(x)\mid g(x)$ in $F[x]$, then either $g(x) = 0$ or $\deg(g(x)) \geq \deg(f(x))$ I'm not really sure how to prove these types of statements
1
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0answers
17 views

Mobius function verification

I am looking to verify my answer to the question $$F(n)=\sum_{d|n}{\mu(d)\sigma(d)}=(-1)^{\omega(n)}\prod_{j=1}^{\omega(n)}{p_j}$$ Where $\mu$ is the Mobius function, $\sigma$ is the sum of divisors ...
1
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2answers
34 views

Diophantine solution to a fraction

How can we find solutions to the following equation: $$ y=\dfrac{x^2-1085}{14718-2x}$$ where $x,\ y$ are integers.
-1
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0answers
33 views

How does this method work? [on hold]

Let $n=16$ for an example: step 1: get set of prims from $1$ to $\sqrt{2n}: \{2, 3, 5\}$, step 2: get set of $n \mod 2, n \mod 3, n \mod 5: \{0, 1, 1\}$, setp 3: from $0$ to $n-3$, ...
1
vote
1answer
21 views

Sizes of Blocks of Consecutive Integers Divisible by at Least One Prime Less than or Equal to $r$.

Let $f(r)$ be the largest integer such that there exists a block of $f(r)$ consecutive integers each divisible by some prime that is less than or equal to $r$. For example, $f(2)=1$ because it is ...
0
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1answer
31 views

regarding pseudo-prime numbers.

If $W$ is an odd composite number and $-1+2^{W-1}$ is divisible by $W$ yet not by $W^2$, then $W^2$ does not divide $-1+ 2^{W(W-1)}$. Is this true? (forgive my use of symbols,I have no good math ...
2
votes
4answers
110 views

What is the remainder when the below number is divided by $100$?

What is the remainder when the below number is divided by $100$? $$ 1^{1} + ...
2
votes
2answers
63 views

Fibonacci number ending with given sequence of digits

Related to this question: For any given sequence of digits, does a Fibonacci number exist ending with such sequence? If not, it would be nice to find the smallest counterexample. (in other ...
2
votes
0answers
25 views

Polynomial Density Theorem

So I was consider Lagrange's 4-square theorem and came up with this generalization: Given a polynomial with rational coefficients $$P(x) = a_0 + a_1x + ... + a_nx^n$$ Determine if there exists ...
0
votes
2answers
65 views

Finding all possible pairs of integers $(a,b)$ such that $a^b=n$. [duplicate]

Given an integer $n$, how can I find all possible pairs of integers $(a,b)$ such that $$a^b=n.$$ A fast algorithm is preferable.
1
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0answers
45 views

How do ramanujan sums and the sum of 2 squares relate?

I read on the Wikipedia page for the ramanujan sum a formula for the sum of 2 squares in terms of the ramanujan sums. But I did not get it. http://en.wikipedia.org/wiki/Ramanujan_sum How do ...
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0answers
65 views

Fermat's Last Theorem 1 Page Proof [on hold]

Fermat’s Last Theorem This theorem basically states that $A^n + B^n \neq C^n$, $n \gt 2$ if $A, B, C$ and $n$ are all positive integers. This inequation can be rewritten as $C^n – B^n \neq A^n$. ...
1
vote
1answer
40 views

Contracted ideals in number fields

I am trying to translate a section of Wolfgang Krull's report "Idealtheorie". At one point (Section $7$ on Quotient Rings) I believe that he makes something like the following statement: Suppose for ...
0
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0answers
20 views

Fermat pseudoprimes p to base 2 (AKA Sarrus or Poulet numbers) with special properties

Are there any known Fermat pseudoprimes $p\;$ to base $2\;$ (Sarrus or Poulet numbers) with the properties $q = (p-1)/2\;$ is prime and $p \equiv 0 \pmod 3?$ I was not able to find any example up to ...
1
vote
1answer
26 views

How find the minimum of the value $n$ such$105\mid \left(9^{p^2}-29^p+n\right)$

Find the minimum of the value $n$,such for any prime number $p>3$,have $$105\mid\left(9^{p^2}-29^p+n\right)$$ My since $$105=5\times 3\times 7$$ so ...
3
votes
0answers
44 views

An Inequality with Prime Numbers

Let $m\geq 8$ be an integer, and let $q$ be the smallest prime that is greater than $m$. Let $p$ be a prime that satisfies $q\leq p<q^2$, and let $p_0$ be the smallest prime such that ...
1
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0answers
43 views

Finding the number of elements in $\left(ℤ[i]\right)_m$

If $m$ = $m_1$ + $m_1i$ ∈ $ℤ[i]$, is there a general formula to find the number of elements in $\left(ℤ[i]\right)_m$?
1
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3answers
236 views

Is there a polynomial equation for $f(n) = n!$ and if so what is it?

And I am not necessarily talking about $f(n) = n(n-1)(n-2)...(3)(2)(1)$ in its factored form; Well it could be that but then I would like a general way of expansion. Thanks in advance!
11
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2answers
1k views

Proof that 26 is the one and only number between square and cube

$x^2 + 1 = z = Y^3 - 1$ Why $z = 26 $ and only $26$ ? Is there an elementary proof of that ?
0
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0answers
23 views

What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
2
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1answer
17 views

A formula occurring in Dirichlet's proof of the infinity of primes in an AP.

While studying Dirichlet's proof of an infinity of primes in any AP with first term and common difference coprime, the formula below involving the gamma function was quoted as being well known. ...
1
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0answers
59 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
1
vote
1answer
29 views

Solution to equation in surd

$\sqrt{a+b\sqrt{c}}=\sqrt{x}+\sqrt{y}$ where $a, b, c\in\mathbb{Z}^+$ and x, y $\in \mathbb{Q} $ Please help show how to disprove or prove. Thanks a lot
0
votes
1answer
15 views

When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
15
votes
3answers
276 views

Equality of Sums

I recently encountered the following equality ($\{\}$ denotes fractional part): $$\sum_{k=1}^{65}k\left\{\frac{8k}{65}\right\}=\sum_{k=1}^{65}k\left\{\frac{18k}{65}\right\}$$ and found it very ...
0
votes
1answer
35 views

Find $x$ such that $o_1^x-o_2^x \neq 2(o_3^x-o_4^x)$ where $o_i$ is an odd number, $o_1>o_2$, $o_3>o_4$ and $x$ is a positive integer

A few hours ago I asked this question. This problem came up while working on a graph labeling problem. I already have a exponential algorithm working. But I want to further reduce the complexity. ...
0
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2answers
35 views

Number theory proof regarding congruences and common divisors

Anyone know how to prove the following statement? If $ a=b $ (mod m) then the common divisors of a,m are the same as those of b,m
0
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0answers
17 views

Given an integer n find smallest integer i such that σ(i)=n. Smallest Inverse Sum of Divisors

Hi All I need some help I am trying to solve this problem which involves computation of sum of divisors and its inverse. In other words Given an integer n find smallest integer i such that σ(i)=n ...
0
votes
1answer
36 views

Number theory proof regarding norms

How would you prove that if $x$ is a prime in $ℤ[i] \Longleftrightarrow$ $N(x)$ is a prime in $ℤ$ N(x) represents the norm of x.
5
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2answers
83 views

Is $2^{3^n}+1$ always divisible by $3^{n+1}$?

Because it certainly seems to be the case for all positiv $n$ but how can I prove it?
1
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0answers
35 views

Prime bounds under RH

Continuing from here, since $$ \sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}=\operatorname{li}(n)-\sum_{k=1}^{\infty}2\ ...
0
votes
2answers
64 views

Legendre symbol of $\left(\!\frac{11}{71}\!\right)$

I'm trying to find the Legendre symbol of $\left(\!\frac{11}{71}\!\right)$ . Here is what I did so far : ...
4
votes
1answer
54 views

Upward continued fractions

Has anybody seen "upward continued fractions", such as $$ \frac{1+\large{\frac{1+\large{\frac{1+...}{a_2}}}{a_1}}}{a_0} \quad? $$ These can be formed, for any real number $x$ with $0<x\le 1$, by ...
0
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2answers
42 views

Number theory proofs regarding units and orders

Hey I just came across this proof but I have no idea how to prove this. If $u$ ∈ $U_m$ has order $n$ and ($k,n$) = 1, then $u^k$ has order n. Any ideas?
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0answers
54 views

About the concept of prime numbers [on hold]

I have developped a new approach of the concept of prime number and I generalized the concept to the reals ! Can you tell me what you think about it, please ? I have already published an article in a ...
1
vote
0answers
79 views

Olympiad number theory question

Let $p,q$ and $r$ be prime numbers. It is given that $p$ divides $qr − 1$, $q$ divides $rp − 1$, and $r$ divides $pq − 1$. Determine all possible values of $pqr$. I think I'm missing something in ...
18
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3answers
297 views
+100

Is this graph connected

Define the following equivalence relation $\sim$: We have $a \ \mathcal{R} \ b$ if and only if $a+b \ | \ ab-1$. $a \sim b$ if and only if there exist a sequence of integers $a_1, \ldots, a_n$ such ...
0
votes
0answers
23 views

Summation of product of Moebius function and arbitary artithmetic function

Suppose $f(n)$ is some arithmetic function and I want to compute effectively the sum $$ \sum_{n \leq x}\mu(n)f(\lfloor\frac{x}{n}\rfloor) $$ where $\mu$ is a Moebius function. Is it possible to ...
0
votes
1answer
45 views

Sort of Binomial Expansion

I was trying to find a general formula for expanding the product: $$\prod_{i=1}^k (a+ib)$$ where $a, b \in \mathbb{R}$. The first few expansions are as follows: $$\prod_{i=1}^1 (a+ib) = a + b$$ ...
2
votes
1answer
51 views

How find the maximum the postive integer $n$ if such $w(n)+\varphi{(n)}=1+\pi{(n)}$

Question: if the postive integer $n$ such $$w(n)+\varphi{(n)}-\pi{(n)}=1$$ find the $\max{(n)}$, where $\varphi{(n)}$ is Euler's totient function (Euler function) $\pi{(n)}$ is ...
0
votes
0answers
19 views

Remainder from polynomial long division

Let $n>1$ and $n\mid p-1$ where $p$ - prime number. Prove that the remainder of dividing $x^p-x$ into $x^n-A$ is equal to $x(A^{\frac{p-1}{n}}-1)$ My sketch proof: We will write this in such form ...
3
votes
3answers
76 views

Diophantine equation abc + abd + acd + bcd= 1

Is there a reference which classifies or at least gives an infinite family of integer solutions to the above equation? A solution to the problem would also be great obviously.
0
votes
0answers
24 views

Infinitely many prime ${2^{(n+1)}}(5^n)−1$ for some Natural number $n$.

How to prove that there exist an infinite number of prime ${2^{(n+1)}}(5^n)−1$ for some Natural number $n$. $$\begin{array}{|c|c|} \hline n & {2^{(n+1)}}(5^n)−1 \\ \hline ...
38
votes
4answers
2k views

Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
3
votes
2answers
28 views

Mobius function summation trouble

I have the following exercise. I am supposed to show that for all natural numbers $n$, that the following equality holds $$\sum_{d|n}{\mu{(d)}d(d)}=(-1)^{\omega{(n)}}$$ Where $\mu$ is the Mobius ...
0
votes
1answer
22 views

integral point on conics

Suppose we have a conic $ax^2 + bxy + cy^2 + dx + ey + f = 0$ where $a,b,c,d,e,f \in \mathbb{Q}$. Is there a way of computing the integer points on this curve. Since it is affine an not projective we ...
3
votes
1answer
46 views

Question on number theory ( related to (Z/p^rZ)* group )

This is a (different version to) question from Serre 'A Course in Arithmetic'.Let p be an odd prime number. $\forall n\geq 1$ (n positive integer), $f$ is defined by: $$f(n)=(-1)^n\prod_{1\le k\le n ...
8
votes
1answer
97 views
+500

An inequality on sequences with each term dividing sum of two neighbouring terms

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is $$\frac{x_{i-1}+x_{i+1}}{x_i} = k_i $$ is an integer for ...