# Is every large enough even integer the sum of two quasiprimes ?

Call a "quasiprime" any integer of the form $$p^{k}$$ with $$k\geq 1$$ and $$p$$ a prime number. In other words, the set of quasiprimes is exactly the set of integers for which the von Mangoldt $$\Lambda$$ function doesn't vanish.

Can it be proven that every sufficiently large even integer is the sum of two quasiprimes ?

• The usual term is simply prime power. – Henning Makholm Oct 1 '18 at 11:11
• Thank you for the information. I thought "prime power" required $k\geq 2$. – Sylvain Julien Oct 1 '18 at 11:15
• Mathworld seems to agree with Wikipedia mathworld.wolfram.com/PrimePower.html but Mathworld brings up $k \geq 2$ much closer to the top. Of course no one is going to vandalize Mathworld between the time I post this and the time you read it. – Robert Soupe Oct 1 '18 at 16:11
• Anyway, for what it's worth, I think it's perfectly reasonable to require $k > 1$. I think it would be clearer to say "primes and prime powers" rather than to try to assign a new meaning to "quasi prime." – Robert Soupe Oct 1 '18 at 16:13

Maybe someone else can say yes for sure, but I can at least say that quite likely yes. If there is a counterexample, it would be an extremely remarkable number.

You were wise to include . If the Goldbach conjecture is true, every even number greater than 2 can be expressed as the sum of two odd primes in at least one way.

What's more, every even number greater than 6 can be expressed as the sum of two distinct primes in at least one way. According to the OEIS, these are the only even nonnegative numbers that can't be expressed as a sum of two distinct primes in two or more ways: $$0, 2, 4, 6, 8, 10, 12, 14, 38.$$

Also allowing nontrivial prime powers (hereafter just "prime powers") helps out a little if we require the summands to be distinct: $$6 = 2^2 + 2$$ $$10 = 2^3 + 2$$ $$12 = 3^2 + 3 = 2^3 + 2^2$$ $$14 = 3^2 + 5$$ $$38 = 3^3 + 11$$

Among the small numbers, primes are more frequent than prime powers, but we can still observe that as the numbers get larger, more options become available.

So while 14 and 38 have fewer options than 12, 140 and 380 both have more than 12, though maybe not as many as 120.

This effect with primes is the Goldbach "comet." It's not strictly increasing, but the running average is. Throwing prime powers in the mix surely fortifies the comet.