All Questions
Tagged with prime-factorization prime-numbers
831
questions
2
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2
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63
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Is there a more efficient way to find the least prime factor?
Assuming $Q_{k} \equiv p_{k}\text{#} + 1$, my goal is to find the least prime factor of $Q_{k}$ for each integer $k = 1 \ldots 100$ . The Python program shown below tries using SymPy to do so, but ...
3
votes
1
answer
199
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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 ...
1
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2
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73
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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
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1
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64
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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 ...
3
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2
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191
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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
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1
answer
58
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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 ...
1
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1
answer
146
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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$ ...
0
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0
answers
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? ...
1
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1
answer
94
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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$ ...
0
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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, $(\...
3
votes
3
answers
221
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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 ...
2
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0
answers
57
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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=\{...
4
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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$ ...
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'...
2
votes
1
answer
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 ...
1
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1
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84
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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 ...
2
votes
1
answer
97
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Show that $n$ has $2^{\omega(n) - 1}$ coprime factor pairs
I am trying to show that $n$ has $2^{\omega(n) - 1}$ coprime factor pairs. I'm pretty sure this is true but I don't see how to prove it. There is no obvious way to use induction.
Here is an example:
$...
2
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0
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62
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Is my proof of the divergence of prime reciprocals valid
I tried to prove the divergence of the prime reciprocals as a challenge and I think I came up with quite an intuitive argument using Borell Cantelli, but maybe not rigorous.
For two primes $p_n>...
2
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0
answers
64
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What percentage of numbers can be written as $n=p*m$ with p prime and $p>m$
What is the chance* that a random positive integer $n$ is the product of a prime $p$ and an integer $m>0$, with $p>m$
Or in other words: when $n$ has a prime factor greater than it's square root
...
3
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1
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217
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A question about prime factorization of composite Mersenne numbers and $(2^p-2)/(2 \cdot p)$
Mersenne numbers are numbers of the form $2^p-1$ where $p$ is a prime number. Some of them are prime for exemple $2^5-1$ or $2^7-1$ and some of them are composite like $2^{11}-1$ or $2^{23}-1$.
I'm ...
2
votes
2
answers
262
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Prime factors of $5^n+6^n+7^n+8^n+9^n+10^n$
I currently run an integer factoring project of the numbers of the form $$5^n+6^n+7^n+8^n+9^n+10^n$$ where $n$ is a non-negative integer.
Do the prime factors have a particular form as it is the case ...
0
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0
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52
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Found a relation regarding the primes, is this interesting?
Define $S_{odd}$ as all $n\in N $ where $n$ is the product of an odd number of distinct primes. Define $S_{even}$ similarly. Thus:
$$S_{odd} = \{2,3,5,...,30,42,....\}$$
$$S_{even} = \{6,10,14, ....,...
1
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2
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92
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SemiPrime Test to determine distance between P and Q
I have two composite primes (semiprimes) where $17*641 = 10897$ and $101*107 = 10807$.
Notice that $10897$ and $10807$ are almost equal. Their square roots are $104.38$ and $103.95$ respectively. But ...
0
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0
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31
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Form of the divisors of a number (Prime Factorization). Is this algorithm-based proof correct? [duplicate]
I am trying to proof the following result:
For a number $n$ whose prime number decomposition is $p_1^{\alpha _1} ... p_m^{\alpha _m}$. Every divisor of $n$ has the form $p_1^{\beta _1} ... p_m^{\beta ...
1
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1
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110
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Open Question: For natural $n$ with prime decomposition $\prod p_i^{r_i}$, define $f(n)=\sum p_i r_i$. Find all $n$ such that $f(n)-f(n+1)=1$.
A question I created myself:
For any $n\in \Bbb{N}$, we can get $n = p_1^{r_1}\cdots p_m^{r_m}$, where $p_i$ are primes. Take $$f(n)=p_1r_1+\cdots+p_mr_m$$
Find all $n$ such that $f(n)-f(n+1)=1$
It ...
4
votes
2
answers
166
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Asymptotics of $p_k$-adic valuation of the sum of the divisors of the $n$-th primorial
Given this product:
$$a(n) = \prod_{k=1}^{n} (1+p_k)$$
where $p_k$ is the $k$-th prime number and which can be interpreted also as the sum of the divisors of the $n$-th primorial (OEIS A054640), is ...
1
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0
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65
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Distribution of the number of prime factors a large number $n$ has?
I know that a large number $n$ has probability $\frac{1}{\ln (n)}$ of having exactly 1 prime factor (i.e. it's prime). But is there any statement on the exact distribution for the number of prime ...
1
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0
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46
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Can the order of a possible further Wieferich prime with respect to base $2$ be prime or a power of two?
A Wieferich prime has the property $$2^{p-1}\equiv 1\mod p^2$$ We only know two Wieferich primes $1093$ and $3511$ , a further Wieferich prime must exceed $2^{64}$.
It is conjectured that there are ...
2
votes
0
answers
50
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The number of integers less than x that have at least two distinct prime factors of bit size greater than one-third the bit size of x
Sander came out with a paper describing how to generate what he calls an RSA-UFO. Anoncoin then utilizes this and mentions that the paper proves that the probability that a randomly generated integer, ...
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1
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157
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Is it correct to say that prime numbers don't exist on $\mathbb{R}$ and $\mathbb{Q}$?
A prime number is defined as: "A non invertible and non zero numer $p$ of a ring $A$ is called a prime number if any time it divides a product of two numbers, it also divides one of the factors&...