Bounded Gaps Between Primes
Author: Julia Stadlmann
Submitted: 31 August 2026
Field: Number Theory
Julia Stadlmann improves the best known unconditional bound for gaps between consecutive primes occurring infinitely often. If
$H1=lim infn→∞(pn+1−pn)$,
then the famous Twin Prime Conjecture predicts $H_1=2$. After Zhang proved in 2013 that $H_1$ is finite, successive improvements by Maynard, Tao and the Polymath8 project eventually established $H_1\le246$. Stadlmann's new result lowers this to
$\boxed{H_1\le240}$.
Thus, there are infinitely many pairs of consecutive primes whose difference is at most $240$. The number $240$ comes from the shortest known admissible $49$-tuple, whereas the previous value $246$ corresponded to an admissible $50$-tuple.
The main innovation is not merely the six-unit improvement but the way it is obtained. Stadlmann combines the classical Bombieri–Vinogradov theorem with newer Zhang-type equidistribution estimates for moduli possessing large smooth factors. Within the GPY/Maynard–Tao sieve framework, this permits a larger region of support for the sieve functions, producing a stronger optimization problem. The proof also develops relaxed equidistribution estimates suited to these more general moduli and uses ideas from Harman's sieve.
The resulting optimization is transformed into a large matrix/eigenvalue problem. Remarkably, the new $240$ bound is obtained using symmetric polynomials only up to degree $21$, whereas Polymath needed degree up to $27$ for the weaker $246$ bound. Direct integration in $49$ variables would be computationally impractical, so Stadlmann develops recursive formulas that reduce the required integrals to matrix multiplication. For $k=49$, the final matrices satisfy the key ratio $>1$, which proves $H_1\le240$. The author presents the result largely as a proof of concept, noting that greater computational resources and higher-degree polynomial bases could potentially push the bound below $240$.
Key takeaways:
ARTICLE
Author: Julia Stadlmann
Submitted: 31 August 2026
Field: Number Theory
Julia Stadlmann improves the best known unconditional bound for gaps between consecutive primes occurring infinitely often. If
$H1=lim infn→∞(pn+1−pn)$,
then the famous Twin Prime Conjecture predicts $H_1=2$. After Zhang proved in 2013 that $H_1$ is finite, successive improvements by Maynard, Tao and the Polymath8 project eventually established $H_1\le246$. Stadlmann's new result lowers this to
$\boxed{H_1\le240}$.
Thus, there are infinitely many pairs of consecutive primes whose difference is at most $240$. The number $240$ comes from the shortest known admissible $49$-tuple, whereas the previous value $246$ corresponded to an admissible $50$-tuple.
The main innovation is not merely the six-unit improvement but the way it is obtained. Stadlmann combines the classical Bombieri–Vinogradov theorem with newer Zhang-type equidistribution estimates for moduli possessing large smooth factors. Within the GPY/Maynard–Tao sieve framework, this permits a larger region of support for the sieve functions, producing a stronger optimization problem. The proof also develops relaxed equidistribution estimates suited to these more general moduli and uses ideas from Harman's sieve.
The resulting optimization is transformed into a large matrix/eigenvalue problem. Remarkably, the new $240$ bound is obtained using symmetric polynomials only up to degree $21$, whereas Polymath needed degree up to $27$ for the weaker $246$ bound. Direct integration in $49$ variables would be computationally impractical, so Stadlmann develops recursive formulas that reduce the required integrals to matrix multiplication. For $k=49$, the final matrices satisfy the key ratio $>1$, which proves $H_1\le240$. The author presents the result largely as a proof of concept, noting that greater computational resources and higher-degree polynomial bases could potentially push the bound below $240$.
Key takeaways:
- The best published bound in this preprint improves from $H_1\le246$ to $H_1\le240$.
- It proves that infinitely many consecutive prime pairs are separated by no more than 240 integers.
- The important advance is the hybrid use of Bombieri–Vinogradov + newer smooth-moduli equidistribution estimates inside the Maynard–Tao sieve.
- The method appears to contain room for further numerical improvement; $240$ is presented as a first demonstration rather than an apparent theoretical limit.
ARTICLE
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