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A conjecture states that if the interval [n-n^θ₀+ε, n] contains primes for all n, then the interval [n-n^θ₁+ε, n] contains primes for almost all n, where θ₁ = 2 θ₀ - 1. In the present paper, the author proves that θ₁ = 1/24, improving the previous exponents 1/21.5 and 1/22 by the author himself. If the conjecture is true, then we can get θ₁ = 1/25.
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Runbo Li (2026) studied this question.
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