What question did this study set out to answer?

To provide a rigorous proof of Legendre's conjecture across all integers using explicit sieve methods.

February 5, 2026Open Access

Unconditional Proof of Legendre's Conjecture

Puntos clave

To provide a rigorous proof of Legendre's conjecture across all integers using explicit sieve methods.
Utilized a stratified sieve method combining linear sieves for primes and semiprimes.
Derived explicit upper and lower bounds on surviving numbers in specific intervals.
Applied classical results in sieve theory and existing estimates for precise calculations.
Proven the existence of at least one prime in designated intervals for integers.
Established explicit bounds on prime gaps supporting Andrica’s conjecture.
Adapted the method to propose new limits on conjectures by Oppermann and Cramér regarding prime distributions.

Resumen

We present a fully rigorous proof of Legendre’s conjecture for all integers using explicit stratified sieve method. By carefully applying a linear sieve at level followed by a second sieve for primes in the range , we obtain explicit lower and upper bounds on surviving numbers and semiprimes in the interval . All constants are fully explicit and derived from classical results in sieve theory, Rosser–Schoenfeld bounds, and Dusart (2010) estimates on . Our method guarantees that at least one prime exists in each interval for , with smaller verified computationally. In addition, this approach yields explicit bounds on prime gaps, providing rigorous support for Andrica’s conjecture, and can be adapted to establish new effective limits for conjectures of Oppermann and Cramér on prime distributions.

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