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 $6k\pm 1$ we can take one of the combinations and write:
$p_1=6k_1+1 ; p_2=6k_2+1 ; p_3=6k_3+1 \Rightarrow p_1+p_2p_3=(6k_1+1)+(6k_2+1)(6k_3+1)=$
$=(6k_4+1)+(6k_5+1)$ , Similarly we can show that for each combination of the $p_1 , p_2 ,p_3$ forms we get $(6k_4\pm 1)+(6k_5\pm 1)$ form. So we have necessary but not sufficient condition for $p_4$ and $p_5$ to be primes. Any idea how to find a sufficient condition for this proof ?