Articles of prime factorization

What is the running time of this algorithm of prime factorization?

I recently figured out my own algorithm to factorize a number given we know it has $2$ distinct prime factors. Let: $$ ab = c$$ Where, $a<b$ Then it isn’t difficult to show that: $$ \frac{c!}{c^a}= \text{integer}$$ In fact, $$ \frac{c!}{c^{a+1}} \neq \text{integer}$$ So the idea is to first asymptotically calculate $c!$ and then keep […]

Express $11!$ and $\binom{23}{11}$ as products of their prime factors

I’m a bit stuck on how to figure this question out without a calculator and what kind of working I’m supposed to show. Any help would be appreciated, thank you. $\ddot\smile$ Factorise $11!$ and $\binom{23}{11}$ into primes without a calculator in any way. Use this to calculate their $\gcd$ and $\rm{lcm}$, and confirm how these […]

A question on polynomials.

Let a polynomial $f\in\mathbb{R}[x,y]$, and $f(x,y)=(x^2+y^2)p(x,y)^2-q(x,y)^2$ and $p,q$ are coprime to each other. When do, $f$ and $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}, $share a non-trivial common factor?

For any prime number $p$ and natural number $i < p$, prove that $p$ divides ${p \choose i}$.

This question already has an answer here: Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$ 5 answers

Consecutive numbers with less than $k$ prime factors?

Let $k$ be an integer. Consider the consecutive numbers with less than $k$ distinct prime factors. Are there arbitary large differences between those numbers ? With other words : Are there arbitary many consecutive numbers having at least $k$ distinct prime factors for every natural number $k$ ? For $k=3$, the jumping champions are : […]

Huge extremely rough provably-composite numbers with no known factors

At Factors of 1000 numbers up to googolplex, it’s shown that for $G$=googolplex and $k \in (57, 101, 143, 167, 173, 219, 231, 257, 279, 303, 387, 587, 719, 741, 789, 813, 941, 971)$, the number $G-k$ has no prime factors under $3.5 \times 10^{14}$. I started wondering if any similarly gigantic numbers could be […]

What can we say about the prime factors of $​^{10}10+23$?

In a video on ultrafinitism I saw a claim that the number $​^{10}10+23$ does not have prime factorization. While I don’t accept the premise of ultrafinitism, I got curious, what can we say about the prime factors of this number? $​^{10}10$ refers to the hyperoperation tetration. In other words, the number is equal to $10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}$, […]

How to efficiently find the largest perfect square dividing a given large integer?

Given a number $n$. I need to find the largest $q$ such that $q^2$ divides $n$. I need the fastest method to find $q$. $q$ can be any number prime or composite. At present I am factorizing the number $n$ to find the highest number $q$. I need a better approach which does not involve […]

Calculation of special natural numbers

As interested in factorization of integers, I had the idea to define the following natural numbers : z(n) := [$\int_n^{n+1} x^x dx$] My questions : 1) PARI can easily calculate z(n) numerically, but for large n it takes quite a long time. Is there an efficient method to calculate z(n) ? (The integral needs to […]

Understanding a plot of composite numbers against the ordinal position of their prime factors

I was toying with primes factors of natural numbers and I have found a graph which caught my interest but one, which I am struggling to understand better. Let us take the composite number $N$=391721. First, find its prime factors which are: 11, 149, 239. Next, note the ordinal number of each prime factors. Here […]