Calculating prime numbers efficiently is a common challenge in programming. In this article, we will explore different approaches to generating prime numbers using SQL Server.

The Problem

A student asked if it was possible to write an SQL statement to generate prime numbers less than a given limit, such as 1000. The challenge was to find a solution that scales well, considering the lack of loops in SQL.

The Solution

There are several useful facts from Number Theory that can help us solve this problem:

1. The prime factors of a number cannot be greater than the square root of that number.
2. All primes are of the form (6 * n ± 1), but not all numbers of that form are primes.

Based on these facts, we can devise different SQL statements to generate prime numbers.

Solution #1

One approach is to load a table with candidate numbers using math fact #2. We can then remove the non-primes by testing if there is a factor among the numbers less than the square root of each candidate.

CREATE TABLE Primes (
  p INTEGER NOT NULL PRIMARY KEY CHECK ( p > 1 )
);

INSERT INTO Primes (p)
(
  SELECT (6 * seq) + 1
  FROM Sequence
  WHERE (6 * seq) + 1 <= 1000
  UNION ALL
  SELECT (6 * seq) - 1
  FROM Sequence
  WHERE (6 * seq) + 1 <= 1000
);

DELETE FROM Primes
WHERE EXISTS (
  SELECT *
  FROM Primes AS P1
  WHERE P1.p <= CEILING(SQRT(Primes.p))
  AND (Primes.p % P1.p) = 0
);

Solution #2

Another approach is to load the candidates into the Primes table by hardwiring the first few known primes into a query. This limits the candidate set and improves performance.

INSERT INTO Primes (p)
SELECT seq
FROM Sequence
WHERE 0 NOT IN (seq % 2, seq % 3, seq % 5, seq % 7, ..);

Solution #3

A different approach is to generate all the non-primes and remove them from the Sequence table.

INSERT INTO Primes (p)
(
  SELECT seq
  FROM Sequence
  WHERE seq <= 1000
)
EXCEPT
(
  SELECT (F1.seq * F2.seq) AS composite_nbr
  FROM Sequence AS F1, Sequence AS F2
  WHERE F1.seq BETWEEN 2 AND CEILING(SQRT(1000))
  AND F2.seq BETWEEN 2 AND CEILING(SQRT(1000))
  AND F1.seq <= F2.seq
  AND (F1.seq * F2.seq) <= 1000
);

Conclusion

In this article, we explored different approaches to generating prime numbers using SQL Server. By leveraging mathematical facts and SQL features, we can efficiently calculate prime numbers up to a given limit. The solutions provided avoid proprietary features and use standard SQL Server features that are compatible across different releases.

Remember, there are more advanced algorithms like the Sieve of Atkin and the various Wheel Sieves that can be used for even better performance. However, the solutions presented here offer a good balance between simplicity and efficiency.