All Questions
Tagged with prime-factorization prime-numbers
830
questions
3
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178
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Smallest "diamond-number" above some power of ten?
Let us call a positive integer $N$ a "diamond-number" if it has the form $p^2q$ with distinct primes $p,q$ with the same number of decimal digits. An example is $N=10^{19}+93815391$. Its ...
- 85.1k
1
vote
2
answers
71
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Expected number of factors of $LCM(1,…,n)$ (particularly, potentially, when $n=8t$)
I’m trying to prove something regarding $8t$-powersmooth numbers (a $k$-powersmooth number $n$ is one for which all prime powers $p^m$ such that $p^m|n$ are such that $p^m\le k$). Essentially, I have ...
- 1,789
1
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Find the next "consecutive-prime composite number" from a given one.
Good day all. I am not a mathematician by a long shot. Please bear with me...
I am playing with "descending-consecutive-prime composite numbers" (I don't think that's the term). These are ...
- 153
3
votes
2
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188
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Convergence of a product involving primes
Let $p_1, ... , p_n, ...$ be the prime numbers in order. Let $n \in \mathbb{N}$ and $q_1, ..., q_n \in \mathbb{N}$. Define
$$
P_n = \prod_{k=1}^n p_k^{q_k} \hspace{1cm} Q_n = \prod_{k=1}^n \left( p_k^{...
- 1,289
1
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1
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56
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Distribution of perfect numbers for a semiprime
Given a semiprime with a length of 120 digits (397bit):
is it possible to meet any assumptions about perfect numbers (prime factors with same length, 199+199bit) for this number?
I have made an ...
- 121
1
vote
1
answer
145
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Minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite
I am looking for references containing results on the minimum $k$ for which every positive integer of the interval $(kn,(k+1)n)$ is composite.
If we denote as $k(n)$ this minimum $k$ for some $n$, $k$ ...
- 1,180
0
votes
0
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39
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Denjoy's Probabilistic Interpretation
Does Denjoy's Probabilistic Interpretation actually "prove" that the Mertens function ratio between numbers with odd number of distinct prime factors and even number of prime factors is 1? ...
- 35
1
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1
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Will there be infinity prime numbers of the sort $a^2 -2$ (where $a$ is odd)?
Will there be infinity prime numbers of the sort $a^2 -2$ (where $a$ is odd)?
To begin with, every odd composite number can be written as $a^2$ or as $a_{x}^2 -a_{y}^2$ as long as either $x$ or $y$ ...
- 1,415
0
votes
0
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67
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Prime number $p$ such that $p+1$ has all given prime numbers as prime factor.
For given finite prime numbers set $P$,
does there exist some prime number $p$ such that
for any $\ell\in P$, $\ell\mid (p+1)$?
For example, if $P=\{2,3,7\}$, then we can take $p=41$.
In this case, $(\...
- 1,934
3
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3
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For what integers $n$ does $\varphi(n)=n-5$?
What I have tried so far: $n$ certainly can't be prime. It also can't be a power of prime as $\varphi(p^k)=p^k-p^{k-1})$ unless it is $25=5^2$. From here on, I am pretty stuck. I tried considering the ...
- 637
2
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56
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What did I get wrong in this Mobius function question? [closed]
$f(n):=\sum\limits_{d\mid n}\mu(d)\cdot d^2,$ where $\mu(n)$ is the Möbius function. Compute $f(192).$
First, I found all of the divisors of 192 by trial division by primes in ascending order:
$D=\{...
- 637
4
votes
0
answers
144
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What are the next primes/semiprimes of the form $\frac{(n-1)^n+1}{n^2}$?
This question is inspired by this question
For an odd positive integer $n$ , define $$f(n):=\frac{(n-1)^n+1}{n^2}$$ as in the linkes question.
For which $n$ is this expression prime , for which $n$ ...
- 85.1k
1
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0
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112
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Plot of the ratios of Goldbach pairs
Preface
I was playing around with matplotlib to generate some number sequences. I wound up looking at Goldbach pairs and manipulating them in different ways. End result was the following plots. I can'...
- 11
2
votes
1
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61
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Asymptotic Approximation towards Sum of the Composite Number's Smallest Prime Factor
I wonder if there is any asymptotic approximation towards the sum of the smallest prime factor of the composite numbers which are less than $n$. This is also the sum of terms whose index is not prime ...
- 21
1
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1
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83
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How many different squares are there which are the product of six different integers from 1 to 10 inclusive?
How many different squares are there which are the product of six different integers from 1 to 10 inclusive?
A similar problem, asking how many different squares are there which are the product of six ...
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