What question did this study set out to answer?

This research aims to refine the GPY sieve and prove bounded gaps between prime numbers.

July 23, 2026Open Access

Maynard's Sieve Refines Prime Gaps to 246, Not Yet Twin Primes — E8 Intelligence Research

Key Points

This research aims to refine the GPY sieve and prove bounded gaps between prime numbers.
Utilized Maynard's theorem to analyze gaps between consecutive primes.
Employed a multidimensional sieve weight to maximize the ratio of sums.
Solved a variational problem involving the Selberg sieve to establish bounds.
Achieved a bound reduction to 246 for gaps between primes.
Showed that infinitely many prime pairs exist with gaps ≤ 246.
Confirmed that the method does not reach the gap of 2 required for twin primes.

Abstract

FINDING: James Maynard refined the GPY sieve to prove bounded gaps between primes, reducing the unconditional bound to 246 and showing infinitely many primes with gap ≤ 246; the method does not yet reach gap 2 (twin primes). | MATH: Let \ (pₙ \) be the nth prime. Maynard's theorem: \ (₍ (p₍+₁ - pₙ) 246\). The method uses a multidimensional sieve weight: \ (w (n) = (₃㶁 | ₍+₇㶁 ₃䃑, , ₃䂵) ² \) with \ (\) chosen to maximize the sum over \ (n\) of \ (₈=₁ᵏ 1₏ (n+hᵢ) \). The optimal constant emerges from solving a variational problem involving the Selberg sieve; the bound 246 comes from the smallest \ (k\) (number of admissible tuples) such that the ratio of sums exceeds 1. For twin primes (gap 2), the required \ (k\) would need to be 2, but the method gives a ratio < 1 for \ (k=2\). | CONNECTION: No direct geometric ratio (0. 382, 0. 618, etc. ) appears. The sieve weights involve lattice structures (mu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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