All Questions
Tagged with prime-factorization elementary-number-theory
608
questions
205
votes
1
answer
24k
views
Are $14$ and $21$ the only "interesting" numbers?
The numbers $14$ and $21$ are quite interesting.
The prime factorisation of $14$ is $2\cdot 7$ and the prime factorisation of $14+1$ is $3\cdot 5$. Note that $3$ is the prime after $2$ and $5$ is the ...
- 4,323
93
votes
2
answers
4k
views
Do most numbers have exactly $3$ prime factors?
In this question I plotted the number of numbers with $n$ prime factors. It appears that the further out on the number line you go, the number of numbers with $3$ prime factors get ahead more and more....
58
votes
2
answers
8k
views
Can I search for factors of $\ (11!)!+11!+1\ $ efficiently?
Is the number $$(11!)!+11!+1$$ a prime number ?
I do not expect that a probable-prime-test is feasible, but if someone actually wants to let it run, this would of course be very nice. The main hope ...
- 85.1k
41
votes
1
answer
2k
views
Is there any other number that has similar properties as $21$?
It's my observation.
Let
$$n=p_1×p_2×p_3×\dots×p_r$$
where $p_i$ are prime factors and
$f$ and $g$ are the functions
$$f(n)=1+2+\dots+n$$
And
$$g(n)=p_1+p_2+\dots+p_r$$
If we put $n=21$ then
$$g(f(21)...
- 2,697
26
votes
1
answer
536
views
Prime factor wanted of the huge number $\sum_{j=1}^{10} j!^{j!}$
What is the smallest prime factor of $$\sum_{j=1}^{10} j!^{j!}$$ ?
Trial :
This number has $23\ 804\ 069$ digits , so if it were prime it would be a record prime. I do not think however that this ...
- 85.1k
22
votes
2
answers
558
views
Is the "cyclotomic diagonalization" always squarefree?
For every integer $n\ge 2$ , define $$f(n):=\Phi_n(n)$$ where $\Phi(n)$ is the $n$ th cyclotomic polynomial. This can be considered as the "cyclotomic diagonalization"
Prove or disprove the ...
- 85.1k
22
votes
3
answers
695
views
Is $\frac{3^n+5^n+7^n}{15}$ only prime if $n$ is prime?
Let $f(n)=3^n+5^n+7^n$
It is easy to show that $\ 15\mid f(n)\ $ if and only if $\ n\ $ is odd.
I searched for prime numbers of the form $g(n):=\frac{3^n+5^n+7^n}{15}$ with odd $n$ and found the ...
- 85.1k
18
votes
2
answers
630
views
$\lim_{n \to \infty}\frac{1}{n}\sum_{k = 1}^nf(k)$ where $f(n)$ is the largest prime factor exponent?
Let $f(n)$ the be largest exponent among exponents of the prime factor of $n$. E.g. $f(80) = 4$ since $80 = 2^4.5$ and the prime factor of $80$ which has the largest exponent is $2$ which occurs with ...
- 20.8k
16
votes
2
answers
2k
views
Are there infinitely many Fermat prime pairs?
Suppose $p$ and $q$ are prime numbers. Let’s call the pair $(p, q)$ a Fermat pair iff $| p - q | < 2\sqrt{2} (pq)^{\frac{1}{4}}$.
Such prime pairs possess a rather interesting property: they can ...
- 15.7k
16
votes
6
answers
16k
views
prime divisor of $3n+2$ proof
I have to prove that any number of the form $3n+2$ has a prime factor of the form $3m+2$. Ive started the proof
I tried saying by the division algorithm the prime factor is either the form 3m,3m+1,3m+...
- 2,935
16
votes
2
answers
609
views
Is there a positive integer $n$ such that the prime divisors of $n^3 - 1$ are $2$, $3$ and $7$?
For a positive integer $k$ write $\pi(k)$ for the set of all prime divisors of $k$. For example, $\pi(24) = \{2,3\}$ and
$\pi(1) = \emptyset$.
Question. Is there a positive integer $n$ for which $\...
- 9,342
15
votes
1
answer
549
views
Largest consecutive integers with no prime factors except $2$, $3$ or $5$?
The number $180$ has a special property. Its prime factors are only $2$, $3$, and $5$. However the number $220$ does not have this special property because one of its prime factors is $11$.
In the ...
- 303
14
votes
1
answer
1k
views
Are all highly composite numbers even?
A highly composite number is a positive integer with more divisors than any smaller positive integer. Are all highly composite numbers even (excluding 1 of course)? I can't find anything about this ...
- 1,552
14
votes
3
answers
909
views
Is there a name/notation for the sum of the powers in a prime factorization
Let $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, where $p_i$'s are distinct primes and $\alpha_i \geq 1$ for all $i$.
Is there any name & notation for the number $\alpha_1 + \alpha_2+ \...
- 547
13
votes
2
answers
1k
views
A question about numbers from Euclid's proof of infinitude of primes
Observe this list:
$$
\begin{aligned}
2+1&=3\\
2\cdot3+1&=7\\
2\cdot3\cdot5+1&=31\\
2\cdot3\cdot5\cdot7+1&=211\\
2\cdot3\cdot5\cdot7\cdot11+1&=2311\\
2\cdot3\cdot5\cdot7\cdot11\...
- 1,673