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.
Description
Legendre's interval property asks for a prime strictly between consecutive integer squares.
Legendre's interval property asks for a prime strictly between consecutive integer squares. The tool reports the actual prime inventory for one supplied interval.1
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.
When to use Legendre Conjecture Checker
Use this tool to 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.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Legendre 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 integerN using the field table and calculation rules, then run the tool.
- Read primesStrictlyBetweenConsecutiveSquares, observedLegendreIntervalNonempty. 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 |
|---|---|---|---|
| integerN | Integer n | number | Required |
Enter integer n from 1 to 999. Sieve through the upper square, remove endpoints and return primes and a nonempty flag.
prime interval = (n²,(n+1)²) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| primesStrictlyBetweenConsecutiveSquares | Primes strictly between consecutive squares | list of number |
| observedLegendreIntervalNonempty | Observed legendre interval nonempty | boolean |
{
"input": {
"integerN": 3
},
"output": {
"primesStrictlyBetweenConsecutiveSquares": [
11,
13
],
"observedLegendreIntervalNonempty": true
}
}Limitations and assumptions
This finite observation is not 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 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.
References
-
Legendre 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.
- 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.