All Questions
Tagged with prime-factorization prime-numbers
831
questions
0
votes
1
answer
141
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
1
vote
1
answer
61
views
How would one show that any given prime p_i must be a factor of some (p_j - 1)? Is that a true property of primes even? [closed]
In short, what I'm asking is, if you were to go through the whole set of positive primes term by term and find for each prime p the prime factorization of (p - 1), whether all prime numbers would ...
- 19
1
vote
1
answer
104
views
Finding the (smallest) next number with the same distinct prime factors as a previous number
(Since there is no answer yet, I removed most "EDIT"'s to make the text more readable)
Today, I was trying to find a natural number $n_{2}$ such that this number has the same distinct prime ...
user1114342
1
vote
1
answer
77
views
Legendre's Conjecture and estimating the minimum count of least prime factors in a range of consecutive integers
I recently asked a question on MathOverflow that got me thinking about Legendre's Conjecture.
Consider a range of consecutive integers defined by $R(x+1,x+n) = x+1, x+2, x+3, \dots, x+n$ with $C(x+1,x+...
- 10.5k
2
votes
2
answers
127
views
The equation $175a + 11ab + bc = abc$ [closed]
Consider all the triples $(a, b, c)$ of prime numbers that satisfy the equation
$$175a + 11ab + bc = abc\ .$$
Compute the sum of all possible values of $c$ in such triples.
I could only get to the ...
- 33
0
votes
2
answers
614
views
Method to finding the number of factors [duplicate]
I've seen that the number of factors of $x$ can be found:
Prime factorising $x$
Taking each power in the factorisation and adding $1$
Multiplying these numbers together.
This results in the number ...
- 855
1
vote
1
answer
118
views
Prove that $\sqrt{-5}$ is a prime in the ring $R=ℤ[\sqrt{-5}]$.
If $R=ℤ[\sqrt{-5}]$ is a ring but not a UFD, prove that the irreducible element $\sqrt{-5}$ is a prime.
This is what I have so far.
Proof: Let $R=ℤ[\sqrt{-5}]$ be a ring but not a UFD. Since $\sqrt{-5}...
0
votes
1
answer
51
views
Difference in two products of prime factorizations
Let $\Phi(n)=\{p_1, p_2, ..., p_k\}$ be the set of prime factors of a number $n$. How does
$$
p_1(n) = \prod_{p_i\in\Phi(n) \\ 1 \le i \le k}{p_i}
$$
compare to
$$
p_2(n) = \prod_{p_i\in\Phi(n) \\ 1 \...
- 128
5
votes
2
answers
663
views
Number of maximal antichains in the set $\{1,2,3,4,5,6,...,120\}$ where the order is by divisibility relation.
Find the number of maximal antichains in the set $\{1,2,3,4,5,6,7,...,120\}$ where the order is divisibility relation. For example, $\{6,7,15\}$ is an antichain but not a maximal antichain, and $\{1\}$...
- 3,800
8
votes
2
answers
223
views
Showing that prime factors of a number is congruent to $1 \pmod 5$
I have come across numbers of the form
$$b=1+10a+50a^2+125a^3+125a^4$$
where $a$ is a positive integer.
Looking at the prime factors of $b$, I am conjecturing that all prime factors of $b$ are $\equiv ...
- 1,106
1
vote
1
answer
61
views
Set of natural numbers related to least common multiple
I have come across the following set in my research, and I am curious whether this has been studied before/if there is a reference for a related construction.
Given a natural number $n$, let $S(n)$ be ...
- 1,261
0
votes
0
answers
51
views
Find upper and/or lower bounds for the least prime $p$ such that $p^n + k$ is the product of $n$ distinct primes
Well, first of all, happy new year to everyone.
I am trying to solve the following problem: "Let $k$ be a fixed natural number. Find the least prime $p$ such that there exists a natural number $...
- 28.7k
5
votes
0
answers
305
views
Valid Elementary Proof of the Bertrand-Chebyshev Theorem/Bertrand's Postulate? [closed]
$\textbf{Theorem}$ (Bertrand-Chebyshev theorem/Bertrand's postulate): For all integers $n\geq 2$, there exists an odd prime number $p\geq 3$ satisfying $n<p<2n$.
$\textit{Proof }$: For $n=2$, we ...
- 445
1
vote
1
answer
56
views
Lets that $p_1,p_2, ...,p_\lambda>2$ be a set of prime numbers. Is there estimation for the summation of $ A=\sum_{i=1}^{\lambda}\varphi(p_i-1)$?
Lets that $p_1,p_2, ...,p_\lambda>2$ be a set of primes number greater than $2$. Is there any exact formula or estimation for the summation
$$
A=\sum_{i=1}^{\lambda}\varphi(p_i-1)
$$
- 775
-1
votes
1
answer
70
views
Suppose a, b are integers and LCM(a, b) = GCD(a, b)^2. What can be said about the prime decompositions of a and b? [duplicate]
Unsure how to approach the problem besides using the fact that the LCM(a,b) * GCD(a,b) = a*b. I see the implication that the GCD(a,b)^3 = a * b. Perhaps it means a and b are different powers of the ...
- 1