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Tagged with prime-factorization divisibility
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Minimum $k$ for which every positive integer of the interval $(kn, (k+1)n)$ is divisible by at least one prime number less than $n$
As a continuation of this question relating the Minimum $k$ for which every positive integer of the interval $(kn, (k+1)n)$ is composite and this other one on the divisibility of numbers in intervals ...
- 1,180
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Divisibility of numbers in intervals of the form $[kn,(k+1)n]$ [duplicate]
I have checked that the following conjecture seems to be true:
There exists no interval of the form $[kn, (k+1)n]$ where each of the integers of the interval is divisible by at least one of the ...
- 1,180
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Conjecture: $\prod\limits_{k=0}^{n}\binom{2n}{k}$ is divisible by $\prod\limits_{k=0}^n\binom{2k}{k}$ only if $n=1,2,5$.
The diagram shows Pascal's triangle down to row $10$.
I noticed that the product of the blue numbers is divisible by the product of the orange numbers. That is (including the bottom centre number $...
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Is there any algorithm better than trial division to factor huge numbers?
Suppose , we want to find prime factors of a huge number $N$ , say $N=3^{3^{3^3}}+2$. We can assume that we can find easily $N\mod p$ for some positive integer $p$ (as it is the case in the example) , ...
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How to get the smallest $n$ that $n^n$ is divisible by $m$.
I have to calculate an integer $n$ when an integer $m$ is given, that $n^n$ is divisible by $m$.
And the thing is, $n$ is the smallest number that satisfies this condition.
Please help me how can I ...
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Given a partial prime factorization of $N$ consisting of all primes $p \leq \sqrt{N}$ that divide $N$, how do I find the rest of the factorization?
Given an integer $N$, let $P$ be the set of all primes less than or equal to $\sqrt{N}$ that divide $N$. Define $P_{prod}$ as $\prod_{p \in P} f_N(p)$ where $f_N(p) \gt 1$ is the largest power of $p$ ...
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Non-squarefree numbers of the form $10^n + 1$
Consider numbers of the form $10^k + 1$. We can look at the prime factorisation of these numbers and note that the smallest such number that has a repeated prime factor is $10^{11} + 1 = 11^2\cdot{}23\...
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Subset of natural numbers with largest amount of divisors
Let $n, k \in \mathbb{N}$, with $k \le n$.
Which $k$ natural numbers not greater than $n$ have the largest amount of divisors altogether?
Formally, let $D(x)$ be the set of positive divisors of some $...
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If $p$ can divide $a^n+b^n+c^n$ , can $p^k$ divide it as well?
Related to this
Is there a method to decide whether a given function of the form $f(n)=a^n+b^n+c^n$ ($a,b,c$ fixed positive integers , $n$ running over the positive integers) satisfies the following ...
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How do we get the number of prime divisors?
We have a natural square-free number $n$ such that $2^5\cdot 3^6\cdot 5^4\equiv 0 \pmod {\tau(n)}$.
Which is the maximum number of different primes that can divide $n$ ?
$$$$
We have that $\tau(n)$ is ...
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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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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
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How to understand special prime factorization method
Normally when we want to find the Prime Factorization of a number, we will keep dividing that number by the smallest prime number (2), until it can't be divided then we move on to the next prime ...
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Divisibility property involving binomial coefficients and largest prime power divisor [duplicate]
Let $p$ be a prime, let $x$ be an integer not divisible by $p$, and let $j\geq 1$. Denote, as usual, by $\nu=\nu_p(j+1)$ the largest exponent such that $p^{\nu}$ divides $j+1$.
My question : is it ...
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How many positive divisors of 7560 are coprime to 15?
I'm trying to find the amount of positive divisors of $7560$ that are coprime to $15$.
I do know how to find the total number of positive divisors a number has but I am not sure how finding those who ...
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