The problem
The sequence \(p_n^{1/n}\), where \(p_n\) is the \(n\)th prime, is strictly decreasing.
Firoozbakht's conjecture
openHistory & significance
Firoozbakht proposed it in 1982 from prime tables: the sequence \(p_n^{1/n}\) falls monotonically, tightening prime-gap control far beyond Legendre-type statements (it implies a strong form of Andrica's conjecture and more). Computational verification has pushed far past any feasible table, yet the conjecture resists proof — it demands global control of prime gaps of a kind no method supplies. If false, the first counterexample lies enormously far out.
Connected problems
- related Andrica's Conjecture
- related Legendre's Conjecture
More in Prime numbers
- same area The Riesel problem
- same area Gilbreath's conjecture
References
- sourcePrime gap
Still open.
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