Twin Prime Counting Function
Count twin-prime pairs whose smaller prime is at most n, testing the upper member even when it exceeds n; the sieve limit is capped at 1,000,000.
Description
The twin-prime counting function counts pairs according to the smaller prime.
The twin-prime counting function counts pairs according to the smaller prime. A pair can therefore qualify even when its larger prime lies above the stated limit.1
The tool returns count. The indexing and output conventions below determine how to read these values.
When to use Twin Prime Counting Function
Use this for the following task: Count twin-prime pairs whose smaller prime is at most n, testing the upper member even when it exceeds n; the sieve limit is capped at 1,000,000. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Twin Prime Counting Function and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter n according to the field table, then run the tool.
- Read count. Use the stated units, indexing and interpretation of the numerical result.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| n | Inclusive limit on the smaller prime (resource limit 1000000) | integer; minimum 0; maximum 1000000 | Required |
Sieve through n, scan possible lower members and test p+2. Use the sieve when the upper member is in range and a direct primality test otherwise.
π₂(n) = count of prime p≤n for which p+2 is prime The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| count | Twin-prime pairs with smaller prime at most n | integer |
{
"input": {
"n": 100
},
"output": {
"count": 8
}
}Limitations and assumptions
The inclusive lower-member limit is capped at one million. This convention differs from counting only pairs wholly contained below n.
Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.
Alternative or Complementary methods
Choose Twin Prime Pair Checker when you need its operation: Check whether two nonnegative integers are prime and differ by two, accepting either order and limiting each candidate to 10¹² for trial division. Choose Prime Counting by the Sieve of Eratosthenes for the complementary operation: Count primes at most n exactly with the sieve of Eratosthenes, excluding zero and one and limiting n to 1,000,000.
Background reference: OEIS A001359.
References
-
OEIS A001359 — OEIS Foundation
Similar or alternative tools
- Twin Prime Pair Checker
Check whether two nonnegative integers are prime and differ by two, accepting either order and limiting each candidate to 10¹² for trial division.
- Prime Counting by the Sieve of Eratosthenes
Count primes at most n exactly with the sieve of Eratosthenes, excluding zero and one and limiting n to 1,000,000.