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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Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$
Are there any useful restrictions to the prime factors of the number
$$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}?$$
The two smallest are $6771419$ and $72153167$, which I found by trial division. The number ...
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Is the quadratic residue the only restriction?
I still search without success for a prime factor of the huge number $$2\uparrow \uparrow 4+3\uparrow \uparrow 4$$
Another way to write this is $$3^{3^{3^3}}+2^{2^{2^2}}=3^{3^{27}}+65536=3^{...
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Quadratic Sieve on the Gaussian Integers
Factoring a large Gaussian integer $z_0 = a+bi$ into Gaussian primes may be done by first factoring the norm $N(z_0) = a^2 + b^2$ over the integers, and then considering the factors of each integer ...
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Numbers between powers of consecutive primes
So if we try to categorize numbers based on the number of their prime factors we would have something as following where $L_n$ is the list of numbers with $n$ prime factors.
$$
L_1 : 2, 3, 5, 7, 11, .....
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Are there infinite many primes $\ p\ $ that cannot divide $\ 3^n+5^n+7^n\ $?
Let $\ M\ $ be the set of the prime numbers $\ p\ $ such that $\ p\nmid 3^n+5^n+7^n\ $ for every positive integer $\ n\ $ , in short the set of the prime numbers that cannot divide $\ 3^n+5^n+7^n\ $.
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Does the following hold as a conjecture for maximum gaps between prime numbers? and can it be proved?
Even though I used matrix related mathjax on the backend, the frontend is intended to be just a regular table.
$$\begin{matrix}
a&X&X:explanation
\\1&1
\\2&3
\\3&5
\\4&(6)&(...
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Prime Factorization With Typos
It is well known that multiplying two large primes to get their product is easy, and factoring the result to get the prime factors is hard, such that if someone has published a large number and says ...
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An open problem concerns primorials plus one having square factors. What about primorials squared plus one?
No primorial plus one has ever been found that has square factors. However, when I started looking at the square of primorials plus one, I found:
(23#)^2 + 1 = 29^2 * 53 * 1116604864937
I've only ...
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least upper bounds that are coprime
Given $n$ natural numbers $p_1$, $p_2$, ... $p_n$ find numbers $q_1$, $q_2$, ... $q_n$ that are pairwise coprimes such that $p_i$ ≤ $q_i$ and such that $\prod_{i=1..n} q_i$ is smallest possible.
I ...
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Do the sequences defined by $a_n=a_{n-1}+(\text{the least prime factor of }a_{n-1})+1$ starting with $2,6,14,\ldots$ merge?
Let $S_k$ be the sequence defined by $a_k(1)=k,\ a_k(n)=a_k(n-1)+(\text{the least prime factor of }a_k(n-1))+1$.
A diagram of these sequences for around $k<100$ is shown below. As you can see, $S_3$...
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Is it possible to "massage" (via shear transformations) a parallelogram with integer-coordinate vertices into an axis-aligned rectangle?
(The problem is my original, unless there's prior art I'm unaware of.)
Given a parallelogram whose vertices have all integer coordinates, you can give it a "massage". Each "move" ...
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Beal's conjecture $A^x+B^y=C^z$
Beal's conjecture is a generalization of Fermat's Last Theorem. It states: If $A^x + B^y = C^z$, where $A, B, C, x, y$ and $z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $...
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Why is multiplication and division easy, yet addition and subtraction horrible when it comes to Prime Factorisations?
If we have a positive integer $n$ and a function $f(n)$, such that $f(n)=3n$ , we can see that the prime factorisation of $n$ simply changes to includes another 3 within its prime factorisation.
E.g $...
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What is the smallest rectangle that could fit all the mentioned rectangles without overlapping?
If there are n rectangles, each with the size 1×2, 2×3, 3×4, 4×5 ... n(n+1), what would be the smallest rectangle in which we could fit all n of these rectangles without any of them being overlapped?
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How to prove: For all $n$, there exist $n$ consecutive positive integers that have different number of prime factors?
We say two positive integers, $m, k\geq 2$, have a different number of prime factors if in their standard form $m=p_1^{k_1}\cdots p_s^{k_s}$ and $k =q_1^{\ell_1}\cdots q_r^{\ell_r}$ it is the case ...
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