Brocard Conjecture Checker
Check local Brocard interval between squares of supplied adjacent primes p≥3, retaining actual prime inventory; nonadjacent endpoints refused, no global conjecture proof.
Description
Brocard's interval property concerns squares of adjacent prime endpoints.
Brocard's interval property concerns squares of adjacent prime endpoints. Choosing merely two increasing primes would change the question.1
Check local Brocard interval between squares of supplied adjacent primes p≥3, retaining actual prime inventory; nonadjacent endpoints refused, no global conjecture proof.
When to use Brocard Conjecture Checker
Use this tool to check local Brocard interval between squares of supplied adjacent primes p≥3, retaining actual prime inventory; nonadjacent endpoints refused, no global conjecture proof.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Brocard Conjecture Checker and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter lowerAdjacentPrime, upperAdjacentPrime using the field table and calculation rules, then run the tool.
- Read actualAdjacentPrimePair, primesStrictlyBetweenPrimeSquares, atLeastFourObservedPrimes. 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 |
|---|---|---|---|
| lowerAdjacentPrime | Lower adjacent prime | number | Required |
| upperAdjacentPrime | Upper adjacent prime | number | Required |
Supply lower prime in [3,997] and upper prime in [5,997], with lower smaller. Verify primality and adjacency using the sieve; reject nonadjacent pairs. Return the actual interior prime list and whether at least four occur.
count primes in (p²,q²), where p and q are adjacent primes Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| actualAdjacentPrimePair | Actual adjacent prime pair | boolean |
| primesStrictlyBetweenPrimeSquares | Primes strictly between prime squares | list of number |
| atLeastFourObservedPrimes | At least four observed primes | boolean |
{
"input": {
"lowerAdjacentPrime": 3,
"upperAdjacentPrime": 5
},
"output": {
"actualAdjacentPrimePair": true,
"primesStrictlyBetweenPrimeSquares": [
11,
13,
17,
19,
23
],
"atLeastFourObservedPrimes": true
}
}Limitations and assumptions
This is a finite interval check without a global conjecture proof.
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. A successfully returned false flag is an evaluated outcome; its meaning is explained above.
Alternative or Complementary methods
Use Oppermann Conjecture Checker: Enumerate exact primes in both open Oppermann intervals (n²,n(n+1)) and (n(n+1),(n+1)²) for supplied n≥2; local observation, not a global proof. For a complementary task, use Legendre Conjecture Checker: Enumerate actual primes in one open consecutive-square interval (n²,(n+1)²); coarser distinct interval operation from Oppermann split, not a global Legendre conjecture proof.
References
-
Brocard Conjecture Checker — tool.io
Similar or alternative tools
- Oppermann Conjecture Checker
Enumerate exact primes in both open Oppermann intervals (n²,n(n+1)) and (n(n+1),(n+1)²) for supplied n≥2; local observation, not a global proof.
- Legendre Conjecture Checker
Enumerate actual primes in one open consecutive-square interval (n²,(n+1)²); coarser distinct interval operation from Oppermann split, not a global Legendre conjecture proof.