# Questions tagged [prime-numbers]

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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### Maximum order of integers coprime to a prime $p$

The following is a lemma I read online, but I don't understand part of the proof. Let $d$ be the maximum possible order among integers $a$ prime to $p$. Then for any integer $a$ not divisible by $p$,...
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### Prove that If $f_n$ where $n>3$ is prime, then $n$ is prime for a Fibonacci series where $f_1$=$f_2$=1

This problem came up in my conversation with a friend—not sure how basic it is, but it seems quite interesting: Prove that if $f_n$ where $n>3$ is prime, then $n$ is prime for a Fibonacci sequence ...
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### Set of prime numbers and subrings of the rationals

Let $P$ denote a set of prime numbers and let $R_{P}$ be the set of all rational numbers such that $p$ does not divides the denominator of elements of $R_{P}$ for every $p \in P$. If $R$ is a subring ...
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### Checking if all elements are prime

I've often come across problems where (as a subproblem) I need to decide whether a list of numbers contains only primes or at least one nonprime. Is there an efficient way to do this? Right now I ...
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### Where can I find a list of sphenic numbers?

According to Wikipedia, A Sphenic number is a positive integer which >is the product of three distinct prime numbers. Anybody knows whether there is a list, say first 1000 sphenic numbers? It ...
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### Estimating number of crossings for Erastothenes' Sieve

In this paper (2.1) I need to understand the formula for the total number of operations: $$\sum_{i=1}^{\pi(\sqrt n)}\frac{n}{p_i} \approx n\ln \ln n + O(n)$$ On a sidenote, since we're only checking ...
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### sums of square free numbers , is this conjecture equivalent to goldbach's conjecture?

As one can notice every integer greater than $1$ is a sum of two squarefree numbers.(numbers that are not divided by some prime square power). Can we prove that? Edit: Can we have bounds for the ...
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### Do there exist any dormant primes?

Suppose that n is an integer > 1 such that: The prime factorization of n is known It is known that (n + 1) is a prime Then: What can be concluded? Among the possibilities are the following: We can ...
Sum of a set of sphenic numbers can't be equal to the sum of any other set of sphenic numbers. By that I meant, Say S is the set of sphenic numbers. Let S$_1$ $\subset$ S. Then there is no such S$_2$ ...