All Questions
Tagged with divisors nt.number-theory
14 questions
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Divisors on product abelian fourfolds
Given a principally polarized abelian surface $A$ with CM of signature $(1,1)$ by an imaginary quadratic number field $K$, I am interested in studying the Néron-Severi group $\text{NS}(A\times A)$. ...
- 91
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161
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Can $P(z)$ have a divisor in a given congruence class?
In the answer to this previous question , Noam D. Elkies proved that for any integer $x$, $x^3-x^2-2x+1$ can only have a divisors equal to $-1$, $0$, or $1$ modulo $7$. I would like to know what is ...
- 7,182
4
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509
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Zsigmondy's Theorem Generalization
Zsigmondy's Theorem states that if $a>b>0$ are coprime integers then for any integer $n\geq 1$ there is a prime $p$ that divides $a^n-b^n$ and does not divide $a^k-b^k$ for any positive integer $...
- 151
2
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Average length of consecutive integers which have an increasing number of divisors
Consider the nine consecutive natural numbers starting from $1584614377$.
...
- 5,470
11
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1
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866
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Is the divisibility graph of the proper divisors of n more often planar than not?
Define the divisibility graph of a set of positive integers as the graph whose vertices are the integers, two of which are joined by an edge if one divides the other.
For all N, is it true that ...
- 3,717
1
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210
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Gcd of linear function
$\DeclareMathOperator\gcd{gcd}$Take $q\in \mathbb N$ and $X>0$ ($q$ not necessarily smaller than $X$). A sum such as
$$\sum_{d\leq X}(q,d)$$
is easily seen to be $\ll q^\epsilon (X+q)$ so that the ...
- 1,381
2
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1
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118
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Effective semi-group of a singular abelian surface
Let $A$ be a singular abelian surface over $\mathbb{C}$; that is, an abelian surface of maximal Picard rank $\rho(A)=4$. By Shioda-Mitani we know $A \cong E \times E'$ where $E,E'$ are isogenous ...
- 1,701
1
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399
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Boundary divisor of projective toroidal compactification
If $F$ is a totally real number field with $[F:\mathbb{Q}] = d>1$, $X$ is the moduli space of Hilbert-Blumenthal Abelian varieties for $F$, and $\overline{X}$ is the projective toroidal ...
- 949
3
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474
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Bound on the number of primitive divisors of the $n$th Fibonacci number
It is a result of Carmichael that for any integer $n > 12$, the Fibonacci number $F_n$ has at least one primitive divisor, that is, a prime factor $p$ such that $p$ does not divide any $F_m$ with $...
user40023
-2
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3
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220
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Is it possible to show that :for $n \geq 1:\sigma(n!-1) $ never be prime and why $\sigma(n!-1)\bmod 10 $ at most is $0$?
This question is related to my question here , I w'd like to check if $n \geq 1:\sigma(n!-1) $ never be prime according to some computations which i did in wolfram alpha to come up with parity of sum ...
user99666
6
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When is $ \sigma(n!-1) $ a perfect square?
I am looking for pairs of positive integers $(m,n)$ such that $ \sigma(n!-1) =m^2$, where $\sigma$ is the sum of divisors function. Examples occur with $(m,n)=(12,5),(1,2)$.
Question: Are there ...
user99666
0
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2
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244
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Is there an example of integers ($x,p, q ,y$ ) which satisfies the below conditions in this claim? [closed]
Edit 01:In order to look divisibility among power divisor function where i would like to know if there a such integer $n>1 $ with y coprime to $x$ then we have: :$\sigma_y(n)\bmod \sigma_x(n)=0$, ...
- 414
7
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861
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Is $n=6$ the only integer satisfies ${\sigma}_x(n) \equiv 0\bmod{n}$ for every odd integer $x > 0$ and $2 (\bmod n)$ if $x$ is even integer? [closed]
After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ i got this result : $\sigma_x(6)=0 \bmod 6$ for $...
- 414
3
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When is the Wendt binomial circulant determinant divisible by 3?
The Wendt binomial circulant determinant $W_n$ can be defined quite simply as a resultant:
$$ W_n = \operatorname{res}(x^n-1, (x+1)^n-1). $$
Truer to its name, one may also define it as the ...
- 4,994