All Questions
Tagged with prime-factorization prime-numbers
831
questions
1
vote
1
answer
84
views
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 ...
- 45
2
votes
1
answer
97
views
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:
$...
- 3,187
2
votes
0
answers
62
views
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>...
- 702
2
votes
0
answers
64
views
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
...
- 702
3
votes
1
answer
217
views
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 ...
- 141
2
votes
2
answers
262
views
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 ...
- 85.1k
0
votes
0
answers
52
views
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:
$$S_{odd} = \{2,3,5,...,30,42,....\}$$
$$S_{even} = \{6,10,14, ....,...
- 702
1
vote
2
answers
92
views
SemiPrime Test to determine distance between P and Q
I have two composite primes (semiprimes) where $17*641 = 10897$ and $101*107 = 10807$.
Notice that $10897$ and $10807$ are almost equal. Their square roots are $104.38$ and $103.95$ respectively. But ...
- 137
0
votes
0
answers
31
views
Form of the divisors of a number (Prime Factorization). Is this algorithm-based proof correct? [duplicate]
I am trying to proof the following result:
For a number $n$ whose prime number decomposition is $p_1^{\alpha _1} ... p_m^{\alpha _m}$. Every divisor of $n$ has the form $p_1^{\beta _1} ... p_m^{\beta ...
- 1,231
1
vote
1
answer
110
views
Open Question: For natural $n$ with prime decomposition $\prod p_i^{r_i}$, define $f(n)=\sum p_i r_i$. Find all $n$ such that $f(n)-f(n+1)=1$.
A question I created myself:
For any $n\in \Bbb{N}$, we can get $n = p_1^{r_1}\cdots p_m^{r_m}$, where $p_i$ are primes. Take $$f(n)=p_1r_1+\cdots+p_mr_m$$
Find all $n$ such that $f(n)-f(n+1)=1$
It ...
- 27
4
votes
2
answers
166
views
Asymptotics of $p_k$-adic valuation of the sum of the divisors of the $n$-th primorial
Given this product:
$$a(n) = \prod_{k=1}^{n} (1+p_k)$$
where $p_k$ is the $k$-th prime number and which can be interpreted also as the sum of the divisors of the $n$-th primorial (OEIS A054640), is ...
- 2,103
1
vote
0
answers
65
views
Distribution of the number of prime factors a large number $n$ has?
I know that a large number $n$ has probability $\frac{1}{\ln (n)}$ of having exactly 1 prime factor (i.e. it's prime). But is there any statement on the exact distribution for the number of prime ...
- 2,230
1
vote
0
answers
46
views
Can the order of a possible further Wieferich prime with respect to base $2$ be prime or a power of two?
A Wieferich prime has the property $$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 ...
- 85.1k
2
votes
0
answers
50
views
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, ...
- 123
1
vote
1
answer
157
views
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&...
- 121