primes Table Function

• primes() – Returns an infinite table with a single prime column (UInt64) that contains prime numbers in ascending order, starting from 2. Use LIMIT (and optionally OFFSET) to restrict the number of rows.

• primes(N) – Returns a table with the single prime column (UInt64) that contains the first N prime numbers, starting from 2.

• primes(N, M) – Returns a table with the single prime column (UInt64) that contains M prime numbers starting at the N-th prime (0-based).

• primes(N, M, S) – Returns a table with the single prime column (UInt64) that contains M prime numbers starting from the N-th prime (0-based) with step S by prime index. The returned primes correspond to indices N, N + S, N + 2S, ..., N + (M - 1)S. S must be >= 1.

This is similar to the system.primes system table.

The following queries are equivalent:

SELECT * FROM primes(10);
SELECT * FROM primes(0, 10);
SELECT * FROM primes() LIMIT 10;
SELECT * FROM system.primes LIMIT 10;
SELECT * FROM system.primes WHERE prime IN (2, 3, 5, 7, 11, 13, 17, 19, 23, 29);

The following queries are also equivalent:

SELECT * FROM primes(10, 10);
SELECT * FROM primes() LIMIT 10 OFFSET 10;
SELECT * FROM system.primes LIMIT 10 OFFSET 10;

Examples {#examples}

The first 10 primes.

SELECT * FROM primes(10);

  ┌─prime─┐
  │     2 │
  │     3 │
  │     5 │
  │     7 │
  │    11 │
  │    13 │
  │    17 │
  │    19 │
  │    23 │
  │    29 │
  └───────┘

The first prime greater than 1e15.

SELECT prime FROM primes() WHERE prime > 1e15 LIMIT 1;

  ┌────────────prime─┐
  │ 1000000000000037 │ -- 1.00 quadrillion
  └──────────────────┘

Solve a modular constraint over primes in a very large range: find the first prime p >= 10^15 such that p modulo 65537 equals 1.

SELECT prime
FROM primes()
WHERE prime >= 1e15
  AND prime % 65537 = 1
LIMIT 1;

 ┌────────────prime─┐
 │ 1000000001218399 │ -- 1.00 quadrillion
 └──────────────────┘

The first 7 Mersenne primes.

SELECT prime
FROM primes()
WHERE bitAnd(prime, prime + 1) = 0
LIMIT 7;

  ┌──prime─┐
  │      3 │
  │      7 │
  │     31 │
  │    127 │
  │   8191 │
  │ 131071 │
  │ 524287 │
  └────────┘

Notes {#notes}

• The fastest forms are the plain range and point-filter queries that use the default step (1), for example, primes(N) or primes() LIMIT N. These forms use an optimized prime generator to compute very large primes efficiently.
• For unbounded sources (primes() / system.primes), simple value filters such as prime BETWEEN ..., prime IN (...), or prime = ... can be applied during generation to restrict the searched value ranges. For example, the following query executes almost instantly:

SELECT sum(prime)
FROM primes()
WHERE prime BETWEEN 1e6 AND 1e6 + 100
   OR prime BETWEEN 1e12 AND 1e12 + 100
   OR prime BETWEEN 1e15 AND 1e15 + 100
   OR prime IN (9999999967, 9999999971, 9999999973)
   OR prime = 1000000000000037;

  ┌───────sum(prime)─┐
  │ 2004010006000641 │ -- 2.00 quadrillion
  └──────────────────┘

1 row in set. Elapsed: 0.090 sec. 

• This value-range optimization does not apply to bounded table functions (primes(N), primes(offset, count[, step])) with WHERE, because those variants define a finite table by prime index and the filter must be evaluated after generating that table to preserve semantics.
• Using a non-zero offset and/or step greater than 1 (primes(offset, count) / primes(offset, count, step)) may be slower because additional primes may need to be generated and skipped internally. If you don't need an offset or step, omit them.