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.

Description

Oppermann's two intervals split the region between consecutive squares at n(n+1).

Oppermann's two intervals split the region between consecutive squares at n(n+1). Checking both open intervals is a finite observation of the stated prime-location property.^[1]

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.

When to use Oppermann Conjecture Checker

Use this tool to 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.

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

How to use Oppermann 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 primesStrictlyBetweenNSquaredAndNNextN, primesStrictlyBetweenNNextNAndNextNSquared, bothObservedOppermannIntervalsNonempty. 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 2 to 999. Sieve through (n+1)², exclude each boundary and return both prime lists and whether both are nonempty.

intervals = (n²,n(n+1)) and (n(n+1),(n+1)²)

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

Output field	Meaning	Value type
primesStrictlyBetweenNSquaredAndNNextN	Primes strictly between n squared and n next n	list of number
primesStrictlyBetweenNNextNAndNextNSquared	Primes strictly between n next n and next n squared	list of number
bothObservedOppermannIntervalsNonempty	Both observed oppermann intervals nonempty	boolean

json

{
  "input": {
    "integerN": 3
  },
  "output": {
    "primesStrictlyBetweenNSquaredAndNNextN": [
      11
    ],
    "primesStrictlyBetweenNNextNAndNextNSquared": [
      13
    ],
    "bothObservedOppermannIntervalsNonempty": true
  }
}

Limitations and assumptions

The result concerns one interval pair and does not provide a global 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 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. 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

Oppermann Conjecture Checker — tool.io

Similar or alternative tools

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.
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.