All Questions
245
questions
3
votes
1
answer
199
views
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
73
views
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
vote
1
answer
94
views
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 $y$ ...
- 1,405
3
votes
3
answers
221
views
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
votes
0
answers
57
views
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
views
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
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
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
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
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
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
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
0
votes
1
answer
140
views
How to factor numbers like 8,023 manually
I was given a random 4-digit number to factor over the prime numbers. My number was 8,023. I tried applying all the divisibility rules up to 36 before giving up on them. I tried using algebra as ...
- 2,370