Firoozbakht Conjecture Checker
Exactly check finite Firoozbakht p_(n+1)^n < p_n^(n+1) using BigInt and true prime indices from complete sieve; observed satisfaction is not global conjecture proof.
Description
Firoozbakht's inequality compares consecutive primes with their actual prime indices.
Firoozbakht's inequality compares consecutive primes with their actual prime indices. Integer-power comparison avoids taking numerically close roots.1
Exactly check finite Firoozbakht p_(n+1)^n < p_n^(n+1) using BigInt and true prime indices from complete sieve; observed satisfaction is not global conjecture proof.
When to use Firoozbakht Conjecture Checker
Use this tool to exactly check finite Firoozbakht p_(n+1)^n < p_n^(n+1) using BigInt and true prime indices from complete sieve; observed satisfaction is not 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 Firoozbakht 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 primeCutoff using the field table and calculation rules, then run the tool.
- Read checkedAdjacentPrimePairCount, exactFiroozbakhtViolationLeftPrimes, allObservedPairsSatisfyInequality. 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 |
Enter integer prime cutoff from 2 to 2000. Sieve from the first prime, compare adjacent pairs using exact BigInt powers and collect left primes where the non-strict opposite inequality holds. Return checked-pair count and whether none violated it.
p_(n+1)^n < p_n^(n+1) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| checkedAdjacentPrimePairCount | Checked adjacent prime pair count | number |
| exactFiroozbakhtViolationLeftPrimes | Exact firoozbakht violation left primes | list of number |
| allObservedPairsSatisfyInequality | All observed pairs satisfy inequality | boolean |
{
"input": {
"primeCutoff": 13
},
"output": {
"checkedAdjacentPrimePairCount": 5,
"exactFiroozbakhtViolationLeftPrimes": [],
"allObservedPairsSatisfyInequality": true
}
}Limitations and assumptions
Zero checked pairs can return true vacuously. Finite satisfaction is not a proof for all primes.
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 Andrica Conjecture Checker: Check finite Andrica √p_next−√p<1 using stable gap/(√p_next+√p); actual prime sieve, no inference about all future pairs. For a complementary task, use Cramér Conjecture Checker: Screen finite adjacent-prime gap/log²(p) ratios under explicit caller coefficient; Cramér conjecture is asymptotic big-O, not a universal coefficient-one theorem proved/disproved by small primes.
References
-
Firoozbakht Conjecture Checker — tool.io
Similar or alternative tools
- Andrica Conjecture Checker
Check finite Andrica √p_next−√p<1 using stable gap/(√p_next+√p); actual prime sieve, no inference about all future pairs.
- Cramér Conjecture Checker
Screen finite adjacent-prime gap/log²(p) ratios under explicit caller coefficient; Cramér conjecture is asymptotic big-O, not a universal coefficient-one theorem proved/disproved by small primes.