Chen’s Theorem states that every sufficiently large even integer can be written as the sum of a prime and a prime or a semiprime (the product of two, not necessarily distinct, primes). Let’s consider the case ‘prime + semiprime’.
$$N = p + P2$$
I’m interested in the semiprime-part (P2) of the theorem. Let’s write P2 as:
$$P2 = p_{i}.p_{j}$$
where $p_{i} \le p_{j} < N$
Is it possible to prove that for every sufficiently large even N there is a solution where $p_{i} > (N/2)^{0.5}$?
I found an interesting paper about Chen’s Theorem with small primes.
Cai, Y.C. Chen's Theorem with Small Primes. Acta Math Sinica 18, 597–604 (2002). https://doi.org/10.1007/s101140200168
Cai proved that there is a solution for $N = p + P2$, for $p \ge N^{0.95}$
‘Small’ primes need ‘large’ semiprimes and large semiprimes can, not necessarily must, be written as $p_{i}.p_{j}$, where $p_{i}$ (and so $p_{j}$) are bigger than $(N/2)^{0.5}$
I don’t know if the ‘small prime’ idea helps. Can anyone help me?