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Define a semiprime to be the product of two (not necessarily distinct) primes, $p_iq_i$.

Conjecture: All squares $\ge 4^2$ are representable as the sum of two distinct semiprimes.


Case 1: Squares of the form $4n^2$, $n \ge 2$.

Suppose $4n^2=p_1q_1+p_2q_2$. Allowing $p_1=p_2=2$ simplifies the relation to $2n^2=\left( q_1+q_2 \right)$. The LHS of the equation is an even number, and the RHS is a sum of two primes, which is equivalent to a weaker version of the Goldbach conjecture.

Case 2: Squares of the form $\left( 2n+1\right)^2$, $n \ge2.$

This is related to a theorem proposed by Meng. Meng proved that every sufficiently large odd integer is representable as the sum of three semiprimes. I am however only interested in the case of odd squares, and whether or not two semiprimes will suffice.


My question is whether or not this conjecture can be proven without assuming Goldbach's conjecture. I have checked squares up to $1519^2 \approx 10^{6.3}$ without finding any counterexamples. I am going to run a few more tests, hopefully up to $10^{10}$. In the meantime, does anyone have any input?

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I just discovered a stronger conjecture that every sufficiently large integer is the sum of two semiprimes. It is apparently suspected to be true. oeis.org/A072966 –  Ryan Sep 3 '13 at 3:03
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I can't help but see the irony in me mulling over this problem. (I'm completely unrelated to the Meng mentioned in Case 2, just for the record) –  Dennis Meng Sep 5 '13 at 21:48

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