Elementary Methods in Number Theory Exercise 1.4.6

Prove that $n,n+2,n+4$ are all primes if and only if $n=3$.

 

 

Proof:$\Rightarrow$:$n$ must be an odd number,let $n=2k+1,k\in\mathbf{N^+}$.So $n,n+2,n+4$ turned into $$2k+1=2(k+2)-3,2k+3=2(k+3)-3,2k+5=2(k+4)-3$$We know that $3$ divides one of $k+2,k+3,k+4$,so $3$ divides one of $2(k+2)-3,2(k+3)-3,2(k+4)-3$,so $2(k+2)-3,2(k+3)-3,2(k+4)-3$ are all primes only in case of  $k=1$ .

 

$\Leftarrow:simple$.