Abstract: This embodiment teaches a variation of GCD-based sieving, building tables of prime products, but intentionally restricting the size of table entries to fit within a single machine word. This combination allows one to mix advantages of the two most popular sieves, while retaining the simple and straightforward structure of the simpler one. Divisor length restriction can provide significant savings in the number of long divisions, but may be implemented with only two very specific primitives. The two primitives offer better optimization capabilities than a fully generic multiword arithmetic library.