Brun Constant Approximation

Compute finite Brun reciprocal sum over actual twin pairs with both primes ≤cutoff; shared prime 5 contributes in both pairs, no extrapolated Brun constant or tail estimate.

Description

Brun's constant is approached by reciprocal contributions from twin-prime pairs.

Brun's constant is approached by reciprocal contributions from twin-prime pairs. Shared primes can appear in more than one distinct pair.^[1]

Compute finite Brun reciprocal sum over actual twin pairs with both primes ≤cutoff; shared prime 5 contributes in both pairs, no extrapolated Brun constant or tail estimate.

When to use Brun Constant Approximation

Use this tool to compute finite Brun reciprocal sum over actual twin pairs with both primes ≤cutoff; shared prime 5 contributes in both pairs, no extrapolated Brun constant or tail estimate.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Brun Constant Approximation and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter primeCutoff using the field table and calculation rules, then run the tool.
Read finiteBrunReciprocalSum, observedTwinPrimePairCount. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
primeCutoff	Prime cutoff	number	Required

Use integer cutoff 0–one million. Sieve actual primes and include pairs only when both endpoints are within the cutoff. Return the sum and pair count; 5 contributes in both (3,5) and (5,7).

finite sum = Σ[twin pairs p,p+2](1/p+1/(p+2))

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
finiteBrunReciprocalSum	Finite brun reciprocal sum	number
observedTwinPrimePairCount	Observed twin prime pair count	number

json

{
  "input": {
    "primeCutoff": 7
  },
  "output": {
    "finiteBrunReciprocalSum": 0.8761904761904762,
    "observedTwinPrimePairCount": 2
  }
}

Limitations and assumptions

No tail extrapolation or certified exact Brun constant is supplied.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Twin Prime Constant Approximation: Compute finite twin-prime constant Euler product over odd primes ≤cutoff using log1p; omitted factors mean finite value is not exact constant or proof of twin-prime infinitude. For a complementary task, use Prime Triplet Checker: Check supplied prime-triplet patterns (p,p+2,p+6) or (p,p+4,p+6), plus exceptional (3,5,7); not any three primes or a twin-pair alias.

References

Brun Constant Approximation — tool.io

Similar or alternative tools

Twin Prime Constant Approximation
Compute finite twin-prime constant Euler product over odd primes ≤cutoff using log1p; omitted factors mean finite value is not exact constant or proof of twin-prime infinitude.
Prime Triplet Checker
Check supplied prime-triplet patterns (p,p+2,p+6) or (p,p+4,p+6), plus exceptional (3,5,7); not any three primes or a twin-pair alias.