Sum of prime and semiprime conjecture

Definitions and mathematical statement

A semiprime is a positive integer with exactly two prime factors counted with multiplicity. Thus it has the form $qr$ with primes $q,r$, and $q=r$ is allowed. An odd semiprime has both prime factors odd. The assertion asks whether \[ \forall N\in2\mathbb Z,\ N>10,\quad \exists\text{ odd primes }p,q,r\quad N=p+qr. \] No distinctness among $p,q,r$ is imposed. For example, the first admissible integer has the representation $12=3+3\cdot3$. The strict lower cutoff excludes $10$, which cannot be expressed using an odd prime at least three and an odd semiprime at least nine.

Every even integer above this cutoff is covered, rather than only all sufficiently large integers or almost all even integers. Replacing the semiprime by a number with at most two prime factors changes the problem, since that would additionally permit a prime in its place. The displayed formula requires exactly two factors counted with multiplicity. A negative answer consists of an even integer $N>10$ for which every odd prime $p<N$ leaves a difference $N-p$ that is not an odd semiprime; a positive answer must cover the entire range.

Short English statement

Is every even integer greater than ten the sum of an odd prime and a product of two odd primes, allowing repeated factors?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3359 (OPG-37404), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Sum of prime and semiprime conjecture. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/sum_of_prime_and_semiprime_conjecture.