All Questions
Tagged with divisors-multiples factorization
10 questions
4
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1
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208
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Representation of a number as a product of $\sqrt{n^2 + 1} + n$
Question. Do there exist two multisets $A, B$ consisting of positive integer numbers such that $|A|$ and $|B|$ have different parity and
$$
\prod_{n\in A}(n + \sqrt{n^2 + 1}) = \prod_{m\in B}(m + \...
- 577
0
votes
0
answers
138
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A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem
I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
3
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0
answers
171
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The kronecker symbol and factorization of $n=\frac{B^N-1}{B-1}$
Let $n=\frac{B^N-1}{B-1}$. Assume $n$ is congruent to 3 modulo 4.
We have the following:
If $N$ is 1 modulo 4, then $N$ is quadratic residue modulo $n$
and $-N$ is quadratic non-residue. The square ...
- 25.4k
1
vote
1
answer
186
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Around a characterization for even perfect numbers, similar than Euclides-Euler theorem, in terms of totatives
In this post we denote the sum of divisors function as $$\sigma(n)=\sum_{1\leq d\mid n}d,$$ then an even perfect number is a positive integer $n\equiv 0\text{ mod }2$ for which $\sigma(n)=2n.$ As ...
2
votes
1
answer
231
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Equations involving arithmetic functions of primorials
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors, $\varphi(n)$ the Euler's totient function and we denote the primorial $\prod_{k=1}^n p_k$ as $N_n$, where $p_k$ denotes the $k$-th prime ...
4
votes
0
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185
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Near Pochhammer symbols: the equation $(n)_m-(k)_l=2$ for integers greater than or equal to two
In this post I consider the following equation involving Pochhammer symbols,
$$(n)_m-(k)_l=2\tag{1}$$
for positive integers $n\geq 2$ and $k\geq 2$, and positive integers $m\geq 2$ and $l\geq 2$.
...
-1
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1
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291
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On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II
I posted this question on MSE two days ago, but did not receive any responses. I have cross-posted it on MO, hoping it gets more attention here and that it is appropriate for this site.
A positive ...
9
votes
2
answers
1k
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runs of consecutive non squarefree integers
This question gained no attention at Math SE.
Call a sequence of $k$ consecutive naturals squary if each one of them is divided by a square > 1. The Chinese Remainder theorem trivially guarantees us ...
- 13.4k
17
votes
0
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420
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Do the coefficients of these irreducible polynomials always become periodic?
Fix $n\in\mathbb N$ and a starting polynomial (or seed) $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_0a_n\ne0$.
Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+...
- 13.4k
7
votes
1
answer
303
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Large gaps between consecutive irreducible polynomials with small heights
For a prime gap of length at least $n$, a trivial upper bound for its first occurrence is $N=n!$ or $N=lcm(2,\dots,n)$. A bit better is $N=p_1\cdots p_n$ where $p_k$ is the $k$th prime, as then $N+2,\...
- 13.4k