# 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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Denote with $p_n$ the $n$-th prime number and let $$h_N(d) = |\{ n : p_{n+1} < N, p_{n+1} - p_n = d \}|$$ be the number of times that prime gap $d$ happens for primes less than $N$. Let $H = \... 0answers 248 views ### When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime? Inspired by Is the number$333{,}333{,}333{,}333{,}333{,}333{,}333{,}333{,}334$a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form$000...0001$... 0answers 319 views ### Understanding Newman's proof of the prime number theorem I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ... 0answers 796 views ### The divergence of the series of reciprocals of primes (proof check): I want to check my attempt at a proof for the divergence of $$\sum_{n=1}^{\infty} \frac{1}{p_n} \tag{ \star }.$$ We begin with assuming that$(\star)$converges. If$(\star)$converges, there is ... 0answers 306 views ### Sophie Germain Star prime number Let us define following prime number : Let$~S_p~$be Sophie Germain Star prime number of the form : $$S_p=12\cdot p \cdot (2p+1)+1,$$ where$p$is a Sophie Germain prime . Note that ... 0answers 1k views ### Sum of odd prime and odd semiprime as sum of two odd primes? How to prove that each sum of odd prime and odd semiprime can be written as sum of two odd primes$(p_1+p_2p_3=p_4+p_5)$? Since we know that each prime number greater than$3$is of the form$6k\pm ...
A number $N$ is given. The object is to find the smallest nonnegative integer $k$, such that $N+k$ is the product of three distinct primes, each having the same number of decimal digits. For example,...