Question

A prime number $p$ is called special if there exist primes $p_1,p_2,p_3,p_4$ such that $p=p_1+p_2=p_3-p_4.$ The number of special primes is

• A. $0$
• B. $1$
• C. more than one but finite
• D. infinite

Answer 1

The correct option is B $1$

Given that $p=p_1+p_2=p_3-p_4$
Apart from $2$ all prime numbers are odd. And we know that Odd + Odd $\ne$ Odd and Odd - Odd $\ne$ Odd. So, one of the part ($p_1$ or $p_2$, $p_3$ or $p_4$) must be $2$

There will be $2$ cases.
Case $1$: $p_1=2=p_3$
$2+p_2=2-p_4$
$\Rightarrow p_2=-p_4,$ which is not possible.

Case $2$: $p_1=2=p_4$
$2+p_2=p_3-2$
$\Rightarrow p_3-p_2=\mathrm{4.....}\left(1\right)$
$p=2+p_2=p_3-\mathrm{2.....}\left(2\right)$

There is only one possibility exist satisfying equation $\left(1\right)$ and $\left(2\right)$, $p_2=3,p_3=7$