Question Number 765 by prakash jain last updated on 09/Mar/15
![List all primes p for which p+2 and p+4 are also primes.](https://www.tinkutara.com/question/Q765.png)
$$\mathrm{List}\:\mathrm{all}\:\mathrm{primes}\:{p}\:\mathrm{for}\:\mathrm{which}\:{p}+\mathrm{2}\:\mathrm{and}\:{p}+\mathrm{4} \\ $$$$\mathrm{are}\:\mathrm{also}\:\mathrm{primes}. \\ $$
Answered by rishabh last updated on 09/Mar/15
![p=3 is the only such number. Let us assume there is a prime p>3. Since p is not divisible by 3, p=3k+1 or 3k+2 where k∈N, k≥1 If p=3k+1 then p+2 is divisible by 3. If p=3k+2 then p+4 is divisible by 3.](https://www.tinkutara.com/question/Q766.png)
$${p}=\mathrm{3}\:\mathrm{is}\:\mathrm{the}\:\mathrm{only}\:\mathrm{such}\:\mathrm{number}. \\ $$$$\mathrm{Let}\:\mathrm{us}\:\mathrm{assume}\:\mathrm{there}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:{p}>\mathrm{3}. \\ $$$$\mathrm{Since}\:{p}\:\mathrm{is}\:\mathrm{not}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3}, \\ $$$$\:\:\:{p}=\mathrm{3}{k}+\mathrm{1}\:\mathrm{or}\:\mathrm{3}{k}+\mathrm{2}\:\mathrm{where}\:{k}\in\mathbb{N},\:{k}\geqslant\mathrm{1} \\ $$$$\mathrm{If}\:{p}=\mathrm{3}{k}+\mathrm{1}\:\mathrm{then}\:{p}+\mathrm{2}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3}. \\ $$$$\mathrm{If}\:{p}=\mathrm{3}{k}+\mathrm{2}\:\mathrm{then}\:\mathrm{p}+\mathrm{4}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{3}. \\ $$