# 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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### More on primes $p=u^2+27v^2$ and roots of unity

Given, $$p=u^2+27v^2=6m+1\tag1$$ and the cubic, $$x^3+x^2-2mx+N=0\tag2$$ with its constant expressed in terms of $(1)$ as, $$N = \frac{1}{27}(1-3p\pm2pu)\tag3$$ and the sign $\pm u$ chosen ...
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### Largest prime factors of two consecutive natural numbers

Let $u(n)$ be the numbers of positive integers $k \le n$ such that the largest prime factor of $k+1$ is greater than the largest prime factor of $k$. Similarly, let $l(n)$ be the numbers of positive ...
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### Are $3$, $9$ the only natural numbers $n$ for which both $2^n-n$ and $2^n+n$ are prime?

I searched for natural numbers $n$, where $2^n-n$ and $2^n+n$ are both prime for the range of $n \le 10^5$ on PARI/GP and found that 3, 9 are the only solutions in this range. Note that since $2^n-n$...
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### A heuristic argument for the Goldbach conjecture?

This question here is purely speculative so be warned if you read on: This question is related to a sequence $b_n$ which is defined here: A series related to prime numbers For the numbers $a_{2n,2}$ ...
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### Iterated Twin Prime conjecture

Here is the beginning of the list of sums of twin prime pairs (OEIS A054735): 8, 12, 24, 36, 60, 84, 120, 144, 204, 216, 276, 300, 360, 384, 396, 456, 480, 540, 564, 624, 696, 840, 864, 924,... "...
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### Primes of the form $a^n-(a-1)^n$

For a given $n$, consider the assertion: $\exists a \in \Bbb Z : a^n-(a-1)^n\ \text{is prime}\tag*{}$ How can one do one of the following: Prove that the assertion is true for all integer $n > 1$...
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### How to approach this sequence? (elementary number theory)

Given is the following sequence: $a_1 = 1$ and $a_n$ equals the biggest prime divisor of $1+ a_1*\dots*a_{n-1}$ . It is then claimed: $11$ does not occur in this sequence. How can one approach this ...
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### Is $p^2+q^2+r^2=3^k$ with primes $p,q,r$ solvable for every odd positive integer $k\ge 3\$?

For the positive odd integers $3\le k\le 25$, the equation $$p^2+q^2+r^2=3^k$$ with primes $p,q,r$ is solvable. Here is one solution for every exponent , calculated with PARI/GP : ...
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### Is the conjecture about prime numbers true?

Let $p_n$ be the $n$-th prime number. Is it true that if $n$ is sufficiently large then will $$p_1×p_2×p_3×...×p_n+1$$ always be a composite number?
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