Questions tagged [prime-factorization]
For questions about factoring elements of rings into primes, or about the specific case of factoring natural numbers into primes.
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Curiosity: $\text{antiprime} = \text{prime} + 1$
The following is just a mathematical curiosity that popped into my head that I thought was interesting. I haven't been able to find anything about it online, although maybe I just am unaware of the ...
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Proving that there is an infinite number of pairs of prime numbers for which $F(n)F(n+1) =pq $ for no $n>1 \in \mathbb{N}$, $F(n)$ is the GPF function
Proving that there is an infinite number of pairs of prime numbers for which $F(n)F(n+1) = pq $ does not hold for any $n>1 \in \mathbb{N}$, $F(n)$ is the GPF function
I have been trying to solve ...
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A conjecture regarding the sum of a number and its prime factorization
I am not sure if this a known problem, but I have a conjecture regarding the prime numbers. Since I don't know much analytic number theory, I thought maybe someone here could prove/disprove it.
Let $n\...
- 2,694
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What is the most frequent largest prime factor of the numbers between two primes?
Let $p_n$ be the $n$-th prime and $l_n, n \ge 2$ be the largest of all the prime factors of the composite numbers between $p_n$ and $p_{n+1}$. Since there are infinitely many prime gaps, each of these ...
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Can we conclude $n=p-1$?
Let $\ n\ $ be a positive integer and $\ p\ $ a prime number such that $$\ p^2\mid (2n)! + n! + 1$$ The only pairs $\ (n,p)\ $ I found so far are
$(1,2)$ , $(2,3)$ , $(10,11)$ , $(106,107)$ , $(4930,...
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Density of Numbers with Exactly One Prime Factor of Multiplicity 1
Let $S$ be the set of positive integers $n$ with the property that exactly one prime factor of $n$ has multiplicity $1$ and every other prime factor has multiplicity greater than $1$ (to be clear, $S$ ...
- 2,726
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Estimation of the number of solutions for the equation $\sigma(\varphi(n))=\sigma(\operatorname{rad}(n))$
For integers $n\geq 1$ in this post we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an ...
user243301
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Prime number theorem for knots?
Is there an analog to the prime-number theorem
describing the distribution of the prime numbers among the integers:
A theorem that describes the distribution of the prime knots,
perhaps with respect ...
- 30.4k
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Is my proof that there are infinite primes actually valid?
I was trying to think of another way of showing that there are an infinite number of primes. I came up with the following argument, but I am not sure if it is valid. I don't know how to make it more ...
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The number of distinct least prime factors in a sequence of consecutive integers
I was thinking about the number of distinct least prime factors in a sequence of consecutive integers and I noticed:
That every $4$ integers, there are $3$ distinct least prime factors. Every $6$ ...
- 10.5k
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Is there a pattern for the distribution of prime factor count for numbers under n?
In the below picture I have charted the distribution of numbers below n by factor count. The bottom line is for all numbers under 100,000 then 200,000 ... all the way to 1,000,000.
They seem to tend ...
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Regularities in a prime-exponent graph
Let $\Omega(n)$ be the number of prime factors of $n$ with multiplicity,
i.e., if $n=p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$,
$\Omega(n) = e_1 + e_2 + \cdots + e_k$ (OEIS).
For example, for $n=9000 = 2^...
- 30.4k
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Conjecture on the prime factorisation of the $\operatorname{lcm}(1,\dots,k) \pm 1$, $k \geq 2$
For $k \in \mathbb{N},\, k \geq 2$, define $S(k) := \operatorname{lcm}(2,\dots,k)$.
$S$ stands for "superprimorial", as it is a name I've seen come up once before and that I kinda like: it ...
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Expected number of digits of the smallest prime factor of $77^{77}-18$
Let $X$ be the number of digits of the smallest prime factor of $$77^{77}-18$$ which is a composite $146$-digit number. ECM indicates that the smallest factor has more than $30$ digits.
Assuming ...
- 85.1k
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Is the conjecture about prime numbers true?
Let $p_n$ be the $n$-th prime number.
Is it true that if $n$ is sufficiently large then will $$p_1×p_2×p_3×...×p_n+1$$ always be a composite number?
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